_ random coding _ is an extremely powerful technique to show the existence of a code satisfying certain properties .it has been used for proving the direct part ( achievability ) of many types of coding theorems .recently , the idea of random coding has also come to be regarded as important from a practical point of view .an ldpc ( low - density parity - check ) code can be constructed by choosing a parity check matrix from an ensemble of sparse matrices . thus , there is a growing interest in randomly generated codes .one of the main difficulties associated with the use of randomly generated codes is the difficulty in evaluating the properties or performance of such codes .for example , it is difficult to evaluate minimum distance , weight distribution , ml decoding performance , etc . for these codes .to overcome this problem , we can take a _probabilistic approach_. in such an approach , we consider an ensemble of parity check matrices : i.e. , probability is assigned to each matrix in the ensemble .a property of a matrix ( e.g. , minimum distance , weight distributions ) can then be regarded as a random variable .it is natural to consider statistics of the random variable such as mean , variance , higher moments and covariance . in some cases, we can show that a property is strongly concentrated around its expectation .such a concentration result justifies the use of the probabilistic approach .recent advances in the analysis of the average weight distributions of ldpc codes , such as those described by litsyn and shevelev , burshtein and miller , richardson and urbanke , show that the probabilistic approach is a useful technique for investigating typical properties of codes and matrices , which are not easy to obtain .furthermore , the second moment analysis of the weight distribution of ldpc codes can be utilized to prove concentration results for weight distributions .the evaluation of the error detection probability of a given code ( or given parity check matrix ) is a classical problem in coding theory , and some results on this topic have been derived from the view point of a probabilistic approach .for example , for a linear code ensemble the inequality , has long been known where is the undetected error probability and is the number of rows of a parity check matrix .since the undetected error probability can be expressed as a linear combination of the weight distribution of a code , there is a natural connection between the expectation of the weight distribution and the expectation of the undetected error probability .  in this paper ,an analysis of the undetected error probability of ensembles of binary matrices of size is presented .an error detection scheme is a crucial part of a feedback error correction scheme such as arq(automatic repeat request ) .detailed knowledge of the error detection performance of a matrix ensemble would be useful for assessing the performance of a feedback error correction scheme .for a given binary parity check matrix , let be the binary linear code of length defined by , namely , where is the galois field with two elements ( the addition over is denoted by ) .the notation denotes the zero vector of length . in this paper , a boldface letter , such as for example ,denotes a binary row vector .  throughout the paper , a binary symmetric channel ( bsc ) with crossover probability ( ) is assumed .we assume the conventional scenario for error detection : a transmitter sends a codeword to a receiver via a bsc with crossover probability .the receiver obtains a received word , where denotes an error vector .the receiver firstly computes the syndrome and then checks whether holds or not .an undetected error event occurs when and .this means that the error vector causes an undetected error event .thus , the undetected error probability can be expressed as where denotes the hamming weight of vector .the above equation can be rewritten as where is defined by .\ ] ] the set is usually called the _ weight distribution _ of .the notation denotes the set of -tuples with weight .the notation  ] if is true ; otherwise , it evaluates to 0 .suppose that is a set of binary matrices .note that may contain some matrices with all elements identical .such matrices should be distinguished as distinct matrices .a probability is associated with each matrix in .thus , can be considered as an _ ensemble _ of binary matrices .let be a real - valued function which depends on .the expectation of with respect to the ensemble is defined by { \stackrel{\triangle}{=}}\sum_{h \in { { \cal g } } } p(h ) f(h).\ ] ] the average weight distribution of a given ensemble is given by . ] where .the notation  ] , dominates the undetected error probability .in such a range , the difference in the average weight distributions corresponding to the random and the bernoulli ensembles is small . on the other hand ,if the crossover probability is small , weight distributions of low weight become the most influential parameter . the difference in the average weight distributions of small weight results in a difference in the error exponent .note that the time complexity of the error detection operation ( multiplication of received vector and a parity check matrix ) is -time for a typical instance of a random ensemble , and is -time for a typical instance of a bernoulli ensemble with constant .a sparse matrix offers almost same error detection performance of a dense matrix with linear time complexity if is sufficiently large .in the previous section , we have seen that the average weight distribution plays an important role in the derivation of average undetected error probability .similarly , we need to examine the _ covariance of weight distribution _ in order to analyze the variance of undetected error probability .the covariance between two real - valued functions defined on an ensemble is given by { \stackrel{\triangle}{=}}e_{{{\cal g } } } [ f g ] - e_{{{\cal g } } } [ f ] e_{{{\cal g } } } [ g].\ ] ]  the next theorem forms the basis of the derivation of the variance of the undetected error probability for the bernoulli ensemble .the covariance of the weight distribution for the bernoulli ensemble is given in the following theorem .[ covsparse ] the covariance of the weight distribution for the bernoulli ensemble is given by for and for where and .+ ( proof ) see appendix .when , becomes the random ensemble .we discuss this case here .we first assume that .let ( i.e. , ) .in such a case , we have .define by the variable takes the following values : substituting into equation ( [ convformula ] ) and using the identity ( [ commutative ] ) , we get another proof of this formula is presented in .   the variance of the undetected error probability is a straightforward consequence of theorem [ covsparse ] .[ spvarthreom ] the variance of the undetected error probability of the bernoulli ensemble , is given by ( proof ) the variance of the undetected error probability is given by \\ & = & e_{{{\cal b}}_{m , n , k}}[p_u^2 ] - e_{{{\cal b}}_{m , n , k}}[p_u]^2 . \end{aligned}\ ] ] we first consider the second moment of the undetected error probability : \\ \nonumber & = & \hspace{-3mm}e_{{{\cal b}}_{m , n , k}}\left [ \left(\sum_{w = 1}^n a_w \epsilon^w ( 1-\epsilon)^{n - w } \right)^2 \right ]  \\ \nonumber & = & \hspace{-3mm}e_{{{\cal b}}_{m , n ,k}}\left [ \sum_{w_1 = 1}^n \sum_{w_2 = 1}^n a_{w_1 } a_{w_2 } \epsilon^{w_1+w_2 }  ( 1-\epsilon)^{2n - w_1-w_2 } \right ] \\ & = &  \hspace{-4mm}\sum_{w_1 = 1}^n \sum_{w_2 = 1}^n \hspace{-1 mm } e_{{{\cal b}}_{m , n , k}}\left[a_{w_1 } a_{w_2}\right ] \epsilon^{w_1+w_2 }  ( 1-\epsilon)^{2n - w_1-w_2}\hspace{-1mm}.\end{aligned}\ ] ] the squared average undetected error probability can be expressed as ^ 2&= & \hspace{-3mm}e_{{{\cal b}}_{m , n , k}}\left [ \left(\sum_{w = 1}^n a_w \epsilon^w ( 1-\epsilon)^{n - w } \right ) \right]^2  \\ \nonumber & = &  \hspace{-4mm}\sum_{w_1 = 1}^n \sum_{w_2 = 1}^n \hspace{-1 mm }  e_{{{\cal b}}_{m , n , k}}\left[a_{w_1}\right ] e_{{{\cal b}}_{m , n , k}}\left[a_{w_2 }  \right ] \\ & \times & \epsilon^{w_1+w_2 } ( 1-\epsilon)^{2n - w_1-w_2}\hspace{-1mm}.\end{aligned}\ ] ] combining these equalities and the covariance of the weight distribution , the variance of undetected error probability is obtained .the covariance of the weight distribution for a given ensemble is useful not only for the evaluation of the variance of .let be a random variable represented by where is a real - valued function of .the covariance of the weight distribution is required more generally for the evaluation of the variance of , which is given by a specialized version ( the case where ) of this equation has been derived in the previous corollary .let us consider the bernoulli ensemble with and .table [ r12 ] displays the weight distributions and undetected error probabilities for the 4 matrices in ..weight distributions and undetected error probabilities [ cols="^,^,^,^,^",options="header " , ]   [ r12 ]  from the definition of a bernoulli ensemble , the following probability is assigned to each matrix : combining the undetected error probabilities presented in table [ r12 ] and the above probability assignment , we immediately have the first and second moments : & = & \frac 2 3 \epsilon - \frac 7 8 \epsilon^2 \\ e_{{{\cal b}}_{1,2,1/2}}[p_u^2 ] & = & \frac { 21}{8 } \epsilon^2 - \frac 3 8 \epsilon^3 + \epsilon^4.\end{aligned}\ ] ] from these moments , the variance can be derived : -e_{{{\cal b}}_{1,2,1/2}}[p_u ] ^2 \\ \label{var12 } & = & \frac 3 8 \epsilon^2 - \frac 3 8 \epsilon^3 + \frac{15}{64 } \epsilon^4.\end{aligned}\ ] ]  we can also consider another route to derive the variance by using corollary [ spvarthreom ] .the covariances of are given by from corollary [ spvarthreom ] , we obtain the variance that is identical to expression ( [ var12 ] ) .  in the case of ( i.e. the case of a random ensemble ) , we can derive a closed form expression for the variance .[ rvartheorem ] for the random ensemble , the variance of the undetected error probability is given by ( proof ) the variance of undetected error probability can be obtained in the following way : -e_{{{\cal r}}_{m , n } } [ p_u]^2 \\ \nonumber & = & \hspace{-3mm}\sum_{w_1 = 1}^n \sum_{w_2 = 1}^n { \rm cov}_{{{\cal r}}_{m , n}}\left[a_{w_1 } , a_{w_2}\right ] \epsilon^{w_1+w_2 }  ( 1-\epsilon)^{2n - w_1-w_2 }  \\ \nonumber & = &  \hspace{-3mm}\sum_{w=1}^n(1 - 2^{-m})2^{-m}{n \choose w } \epsilon^{2w } ( 1-\epsilon)^{2n - 2w } .\end{aligned}\ ] ] the second equality is due to corollary [ spvarthreom ] .the last equality are due to eq .( [ covrandom ] ) .we can further simplify the expression using the binomial theorem : the last equality is the claim of the theorem .the next example facilitates an understanding of how the average and the variance of behave .we consider the random ensemble with , and the bernoulli ensemble with ( labeled `` sparse '' in fig . [ fig - curve1 ] ) .figure [ fig - curve1 ] depicts the average undetected error probabilities of the two ensembles .it can be observed that the average undetected error probability of the random ensemble monotonically decreases as decreases .in contrast , the curve for the bernoulli ensemble has a peak around .+ random ensemble : , sparse matrix ensemble : .figure [ fig - curve2 ] shows the variance of for the above two ensembles .the two curves have a similar shape , but the variance of the sparse ensemble is always larger than that of the random ensemble .+ random ensemble : , sparse matrix ensemble : .we here discuss the asymptotic behavior of the covariance of the weight distribution and the variance of for the bernoulli ensemble .the following corollary explains the asymptotic behavior of the covariance of the weight distribution .[ asymptcov ] let the asymptotic growth rate of the covariance of the weigh distribution of the bernoulli ensemble be defined by for and .the asymptotic growth rate is given by for and for where is defined by the function is defined by ( proof ) we here rewrite the covariance formula ( [ convformula ] ) into asymptotic form . by using the binomial theorem , we have by using this identity , the covariance in ( [ convformula ] ) can be rewritten in the following form : where is defined by   letting , we have and if is a constant and , then , making use of the identity  we get combining these asymptotic expressions , the claim of the corollary is derived .the following corollary gives the asymptotic growth rate of the variance of the undetected error probability .the asymptotic growth rate of the variance of the undetected error is given by where is given by ( proof ) it is evident that holds .combining this identity and corollaries [ spvarthreom ] and [ asymptcov ] , we immediately have the claim of the corollary .the second moment of the weight distribution for a given ensemble is given by \\\nonumber  & = &  e_{{{\cal g}}}\left [   \sum _ { { \mbox{\boldmath } } \in z^{(n , w_1 ) } } \sum _ { { \mbox{\boldmath } } \in z^{(n , w_2 ) } }   i[h { \mbox{\boldmath } } ^t = 0^m ] i[h { \mbox{\boldmath } } ^t = 0^m ] \right].\end{aligned}\ ] ] for . since i[h { \mbox{\boldmath } } ^t = 0^m ] = i[h { \mbox{\boldmath } } ^t = 0^m , h { \mbox{\boldmath } } ^t = 0^m],\ ] ] we have  \\\nonumber  & = & \hspace{-3 mm }  e_{{{\cal g}}}\left [   \sum _ { { \mbox{\boldmath } } \in z^{(n , w_1 ) } } \sum _ { { \mbox{\boldmath } } \in z^{(n , w_2 ) } }   i[h { \mbox{\boldmath } } ^t = 0^m , h { \mbox{\boldmath } } ^t = 0^m ] \right ] \\\label{secmom }  & = & \hspace{-7 mm }   \sum _ { { \mbox{\boldmath } } \in z^{(n , w_1 ) } } \sum _ { { \mbox{\boldmath } } \in z^{(n , w_2 ) } }   e_{{{\cal g}}}\left [   i[h { \mbox{\boldmath } } ^t = 0^m , h { \mbox{\boldmath } } ^t = 0^m ] \right].\end{aligned}\ ] ]  we here encounter a problem of evaluating probability of occurrence of both and . in preparation to solve this problem , we will introduce some notation :  for a given pair , the index sets are defined as follows : : x_k = 1 , y_k = 0 \ } \\ i_2 & { \stackrel{\triangle}{= } } & \{k \in [ 1,n ] : x_k = 1 , y_k = 1 \ } \\ i_3 & { \stackrel{\triangle}{= } } & \{k \in [ 1,n ] : x_k = 0 , y_k = 1 \ } \\ i_4 & { \stackrel{\triangle}{= } } & \{k \in [ 1,n ] : x_k = 0 , y_k = 0 \},\end{aligned}\ ] ] where and these regions are illustrated in fig.[fig - regions ] .the size of each index set is denoted by .let be a binary -tuple .the partial weight of corresponding to an index set is denoted by , namely   .,title="fig : " ] +  since the index sets are mutually exclusive , the equation holds and can take an integer value in the following range : the size of each index set can be expressed as , , .let and be binary vectors satisfying . in this proof , we first prove the following equality : ]   \\ \label{xyprob } & = & \left(\frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2  - 2 v } } { 4 } \right)^m\end{aligned}\ ] ] where , and .the support set is defined by : v_i \ne 0 \},\ ] ] where .we need to consider the following three cases : case ( i ) : ( i.e. , the intersection of and is not empty but does not include ) , case ( ii ) : ( i.e. , the intersection of and is empty ) , case ( iii ) : ( i.e. , includes ) .  we first study case ( i ) .suppose that a binary -tuple is generated from a bernoulli source with = p ( i \in [ 1,n]) ] takes 1 .the probability that has an odd weight is given by .for example , the probability that becomes even is where .based on the above argument , we can write the probability  ] under the condition is thus given by  \\ \nonumber & = & \left(\frac{1+z^{i_2}}{2}\right ) \left(\frac{1 + z^{i_3}}{2 } \right ) \\\nonumber & = & \frac{1+z^{w_1}+z^{w_2 }  + z^{w_2 - w_1 }  } { 4 } \\ & = & \frac{1+z^{w_1}+z^{w_2 }  + z^{w_1 + w_2 - 2 v }  } { 4}. \end{aligned}\ ] ] we next consider the case .for this case , we also have  \\ \nonumber & = & \frac{1+x^{w_1}}{2 } \\ & = & \frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2  - 2 v } } { 4}.\end{aligned}\ ] ]  in summary , for any cases ( cases ( i ) ,( ii ) , ( iii ) ) ,  = \frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2  - 2 v } } { 4}\ ] ] holds . since the rows of parity check matrices in can be independently chosen , we obtain eq .( [ xyprob ] ) in the following way : ] \\\nonumber & = & pr[h { \mbox{\boldmath } } ^t = 0 , h { \mbox{\boldmath } } ^t = 0 ]  \\ \nonumber & = & pr [ { \mbox{\boldmath } } { \mbox{\boldmath } } ^t = 0 , { \mbox{\boldmath } } { \mbox{\boldmath } } ^t = 0]^m \\  & = & \left(\frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2  - 2 v } } { 4 } \right)^m.\end{aligned}\ ] ]  combining ( [ secmom ] ) and ( [ xyprob ] ) , we have  \\\nonumber  & = & \hspace{-7 mm }   \sum _ { { \mbox{\boldmath } } \in z^{(n , w_1 ) } } \sum _ { { \mbox{\boldmath } } \in z^{(n , w_2 ) } }   e_{{{{\cal b}}}_{n , m , k}}\left [   i[h { \mbox{\boldmath } } ^t = 0^m , h { \mbox{\boldmath } } ^t = 0^m ] \right ] \\\nonumber  & = & \hspace{-7 mm }   \sum _ { { \mbox{\boldmath } } \in z^{(n , w_1 ) } } \sum _ { { \mbox{\boldmath } } \in z^{(n , w_2 ) } } \left(\frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2  - 2 v } } { 4 } \right)^m \\ \nonumber & = & \hspace{-2mm}\sum_{v= \max\{0,w_1+w_2 - n\}}^{w_1 } { n \choose w_1 } { w_1 \choose v } { n - w_1 \choose w_2 - v } \\\label{2nd } & \times & \left(\frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2  - 2 v } } { 4 } \right)^m.\end{aligned}\ ] ] since  = { n \choose w } \left(\frac{1+z^w}{2 } \right)^m\ ] ] holds , we thus have e_{{{{\cal b}}}_{n , m , k}}\left[a_{w_2 } \right ] \\  \nonumber & = & { n \choose w_1}{n \choose w_2 } \left(\frac{1+z^{w_1}}{2 } \right)^m \left(\frac{1+z^{w_2}}{2 } \right)^m \\ \nonumber & = & \sum_{v= \max\{0,w_1+w_2 - n\}}^{w_1 } { n \choose w_1 } { w_1 \choose v } { n - w_1 \choose w_2 - v } \\  \label{exp2 } & \times & \left(\frac{1+z^{w_1 } + z^{w_2 } + z^{w_1 + w_2 } } { 4 } \right)^m .\end{aligned}\ ] ] the last equality is due to the following combinatorial identity : we are ready to derive the covariance of weight distributions for the case . substituting ( [ 2nd ] ) and ( [ exp2 ] ) into -  e_{{{{\cal b}}}_{n , m , k}}\left[a_{w_1 } \right ] e_{{{{\cal b}}}_{n , m , k}}\left[a_{w_2 } \right],\end{aligned}\ ] ] we have ( [ convformula ] ) in the claim part of the theorem .since the definition of covariance is commutative , holds if .this work was partly supported by the ministry of education , science , sports and culture , japan , grant - in - aid for scientific research on priority areas ( deepening and expansion of statistical informatics ) 180790091 .

Abstract: in this paper , an analysis of the undetected error probability of ensembles of binary matrices is presented . the ensemble called the _ bernoulli ensemble _ whose members are considered as matrices generated from i.i.d . bernoulli source is mainly considered here . the main contributions of this work are ( i ) derivation of the error exponent of the average undetected error probability and ( ii ) closed form expressions for the variance of the undetected error probability . it is shown that the behavior of the exponent for a sparse ensemble is somewhat different from that for a dense ensemble . furthermore , as a byproduct of the proof of the variance formula , simple covariance formula of the weight distribution is derived .


arrival processes provide a useful description for events occurring with some probability in a given time or space interval .applications range from communications and transports to medicine ( e.g. , diagnostic imaging ) and astronomy ( e.g. , particle detection ) , . from a statistical point of view the prominent model for arrival processesis notoriously poisson . indeed , even when independence between arrivals can not be assumed , the superposition of a large number of non - poisson processes is approximately distributed as a poisson ( palm - khintchine theorem ) , .also , the limit distribution of counting processes described by the binomial distribution is a poisson , according to the law of rare events .  in modern network contexts ( as , e.g. , data , communication and sensor networks or intelligent transportation systems ) estimating the process intensity at different locations , i.e. , the arrival rates at different nodes , is an important preliminary problem to be solved in order to gain context awareness .clearly , each arrival rate can be estimated by performing a _decentralized _ maximum likelihood ( ml ) estimation based only on the local arrivals at the given node . in this paper we want to investigate how the estimation can be improved by exploiting the cooperation among the nodes .recently , a great interest has been devoted to cooperation schemes in which estimation is performed by a network of computing nodes in a distributed way , rather than by collecting all the data in a central unit , . how to use the information from other nodes in the network and how to design a distributed algorithm merging such information will be the focus of the paper .distributed estimation has received a widespread attention in the distributed computation literature , especially as a natural application of linear ( average ) consensus algorithms , see , e.g. , or the recent surveys .nodes typically interact iteratively with their neighbors by means of a `` diffusion - like '' process in which the estimation is improved by suitably combining the estimations from neighboring nodes , . an incremental and a diffusive distributed algorithm with finite time convergenceare proposed in for ( static ) state estimation .distributed optimization is also strictly related to distributed estimation . in distributed alternating direction method of multipliers ( admm ) has been introduced as a tool for distributed ml estimation of vector parameters in a wireless sensor network . notice that admm and other distributed optimization algorithms , as , e.g. , , can be used as building blocks to solve parts of an estimation problem in a distributed set - up .ml approaches for distributed estimation of a commonly observed parameter have been proposed also in . in consensus - based algorithmshave been developed to simultaneously estimate a common parameter measured by noisy sensors and classify sensor types .in consensus - based algorithms have been developed to estimate global parameters in a linear bayesian framework . in a distributed algorithm is proposed for adaptive bayesian estimation of a common parameter with known prior .dynamic methods have been also proposed in which the nodes keep collecting new measurements while interacting with each other . in a diffusion - based recursive least squares ( rls ) algorithm is proposed to estimate a constant parameter , but with dynamically acquired measurements .finally , in and distributed admm - based algorithms are proposed for the estimation of random signals and dynamical processes .  differently from the above references , in our work we consider a more general bayesian framework that allows the nodes to improve their local estimate , rather than reaching a consensus on a common parameter .as detailed later , we will consider a special model for continuous mixtures of poisson variables .poisson mixtures have been widely used in the arrival - rate estimation literature to model non - homogeneous scenarios , see , e.g. , as early references .the survey provides an extensive review of properties and applications for poisson mixtures . in a bayesian setting the classical mixing ( prior ) distribution is the gamma , which results in a closed - form posterior .the main contribution of the paper is twofold .first , we propose a distributed estimation scheme for arrival rates in an asynchronous network , based on a hierarchical probabilistic framework . specifically , we develop an empirical bayes approach , in which the arrival rates are treated as random variables , whose prior distribution is parametrized by an unknown hyperparameter to be determined via ml estimation . in particular , we borrow from the centralized statistics literature the classical gamma - poisson model ( for poisson mixtures ) , assuming however the hyperparameter is unknown , and extend it to a scenario with non - homogeneous sample sizes .we show that the ml estimation of the hyperparameter is a separable optimization problem , that can be solved in a distributed way over the network by using a distributed optimization algorithm .thanks to this modeling idea , the local estimates are obtained by taking advantage of the whole network data , thus improving the accuracy especially when the amount of local data is scarce . with this approach we are able to capture the fact that arrival rates are not the outcomes of isolated phenomena , but rather the expression of global properties of the process . for this `` optimal '' estimatorwe characterize mean and variance at steady - state ( i.e. , once agents have reached consensus on the optimal solution ) for networks with a large number of agents .second , we propose an alternative ad - hoc distributed estimator that , although suboptimal , performs comparably to the optimal one .the main advantage of this ad - hoc estimator is that the resulting algorithm has a simple update rule based on linear consensus protocols , thus exhibiting the same appealing exponential convergence . for the ad - hoc estimatorwe also characterize transient ( at any time - instant ) mean and variance for a given number of agents .notably , we show that at steady - state and for large number of agents the ad - hoc estimator attains the performance of the optimal one .  to strengthen these two contributions we perform a monte carlo analysis and compare the theoretical expressions obtained for mean and variance with their sample counterparts .the numerical computations confirm the theoretical analysis and show that some key assumptions made for a rigorous , but tractable analysis are not restrictive .moreover , they highlight other interesting features of the proposed distributed estimators .for example , the ad - hoc estimator achieves performances close to the optimal one even for a limited number of nodes .the paper is organized as follows . in section [ sec : prob_form ] we introduce the model of a monitoring network and set up the estimation problem . in section [ sec : estimators ] we develop the proposed distributed estimators based on the empirical bayes approach .the statistical performances of the two estimators are analyzed in section [ sec : theoretical_analysis ] , while in section [ sec : numerical_analysis ] we perform a monte carlo analysis to confirm the theoretical results .we consider a _ network of monitors _ having _ sensing _ , _ communication _ and _ computation _ capabilities .that is , each monitor can measure the number of arrivals in a given measurement time - scale ( e.g. , 1 second or 1 minute ) , share local data with neighboring agents , and perform local computations on its own and its neighbors data .the objective is to fuse the data in order to improve the estimation of the arrival rates .formally , each node collects measurements _ asynchronously _ in an _ observation window _ , over which the underlying process can be assumed to be stationary , i.e. , the set of rates can be considered constant over the observation window .clearly , the arrival rates have stationary increments , so that the number of arrivals in disjoint intervals are statistically independent .a measurement consists of the number of arrivals detected at a given location in the ( common ) _ time - scale interval_. accordingly , for each monitor we introduce the following variables :  * unknown arrival rate ; * the -th collected measurement ( number of arrivals per time - scale interval ) ; *  ] .the marginal distribution of agent is derived from the joint distribution of and , by using eq . into eq .the likelihood is rewritten as : thus , the ml estimator of can be found by solving the following optimization problem   the problem can be solved in closed - form only for the _ homogeneous _ case where all are equal , i.e. , for , recalling that is the total number of measurements . in this casethe ml estimator of based on the entire set of measurements is given by where , and .after obtaining an estimate for , the empirical bayes estimator of the arrival rate that minimizes the mean square error ( mmse ) can be obtained by computing the conditional mean of the posterior distribution .the latter is given by the ratio between the joint pdf and the marginal pdf as from eq ., i.e. , eq . is a gamma pdf with parameters , hence the _ empirical bayes mmse estimator _ of each is = \frac{{\hat{b}^{\text{ml}}}}{{\hat{b}^{\text{ml}}}n_i+1 } ( a + \sigma_i).\ ] ]  it is worth highlighting that the bayesian estimate is especially useful when is small .in fact , nodes improve the quality of their local estimate by combining the frequentist estimation ( based only on local observations ) with a correction term ( based on a prior global knowledge ) , which is estimated in a cooperative way .indeed , we can rewrite , with .when the local information is abundant ( and thus ) , the mmse estimator tends towards , meaning that when is large no further information can be inferred from the network .conversely , when local information is scarce , i.e. , the sample size is small , even one , the mmse estimator approaches the estimate of the global mean = a b ] and ^t ] , ^t { \stackrel{\textrm{\tiny def}}{=}}{\bm{n}} ] .it is worth noting that the column stochasticity assumption above is not the usual assumption used in linear consensus algorithms in which row stochasticity is assumed .[ lem : weak_ergodicity ] let be a uniformly jointly strongly connected graph ( assumption [ assum : graph ] ) and a sequence of matrices satisfying assumption [ assum : col_stochastic ] . then denoting with element , the following holds true .  1 .the matrix sequence is weakly ergodic , i.e. , for all .2 .  there exists some such that for all , , i.e. , for any ,   the result is well - known and can be found , e.g. , in .further references on this result under the same or different connectivity assumptions are .[ prop : ad_hoc_convergence ] let assumption [ assum : graph ] hold .then the ad - hoc distributed estimator ( algorithm [ alg : ad - hoc_estimator ] ) satisfies   from lemma [ lem : weak_ergodicity ] , is weakly ergodic so that there exists a sequence of stochastic vectors  ] is irreducible .the result can be proven by following the same line of proof developed therein .thus , the ad - hoc estimator could be implemented also in a monitoring network with stochastic gossip communication .the proposed algorithms are based on the assumption that the arrival - rates are constant in the observation window .therefore , one can apply the algorithms iteratively by recomputing the estimates on different windows of data , thus getting time - varying arrival - rates .  to conclude this section ,we point out that the update of the subgradient push optimization algorithm includes a push - sum consensus step , i.e. , a diffusive update based on a column stochastic matrix ( with coefficients for ) , which is the same used in the ad - hoc distributed estimator ( algorithm [ alg : ad - hoc_estimator ] ) .however , in the subgradient push this update is part of a gradient descent step .in fact , the role and the evolution of the involved variables , i.e. , and respectively , are different as well as the convergence rates . indeed , the ad - hoc estimator exhibits the exponential convergence of linear consensus protocols as opposed to the much slower rate of the subgradient - push .these considerations , together with the lower computational burden , make the ad - hoc distributed estimator appealing even though not optimal . in the following section we will show that it actually performs very closely to the mmse estimator .in this section we analyze the performance of the proposed distributed estimators . in particular , for the empirical bayes distributed estimator , lacking a closed form for the update rule of and , in turn , of , we are able to derive only steady - state ( ) and asymptotic ( ) bounds .conversely , for the ad - hoc distributed estimator a transient analysis ( at any ) can be derived for any .  to develop the performance analysis, we consider a `` special '' agent , we label it as , that does not participate to the computation of ( respectively ) . under this assumption , it turns out that is independent of ( respectively ) .clearly , is also independent of any local estimate ( respectively ) , , at any .consistently , here we can simply assume that for the computation of ( respectively ) , agent uses the local estimate of computed by one of its in - neighbors , that is , e.g. , for some .notice that , in practice , the analysis developed for such an agent holds approximately for any node in the network participating to the distributed computation .indeed , due to the large number of agents , for any agent ( running the distributed algorithm ) , ( respectively ) and are very weakly correlated .the validity of this statement will be corroborated by the monte carlo analysis in the next section .we start by deriving the cramer - rao lower bound ( crb ) for any unbiased estimator of , i.e. , \geq \mathrm{crb}(b).\ ] ]  the crb for the estimation of the hyperparameter is given by [ lem : crb ]  the proof is reported in appendix [ sec : lem_crb ] .   for the empirical bayes distributed estimator we can analyze the performance only once consensus on the optimal value , and thus on , has been reached .specifically , let us recall that   despite no analytical expression is available for the ml estimator ( nor for its moments ) the asymptotic analysis ( ) of the corresponding mmse estimator can be obtained by using the properties of ml estimation . in particular ,any ml estimator is asymptotically unbiased and efficient , hence \rightarrow b ] as . from equation , so that \rightarrow 0 ] . for tractabilitywe will consider respectively the second - order and the first - order approximations for mean and variance .the numerical analysis in section [ sec : numerical_analysis ] will confirm the validity of such an approximation .[ thm : mean_var_eb_estimator ] consider a network of monitors as in section [ sec : prob_form ] running the empirical bayes distributed estimator .then , as , it holds true \longrightarrow \frac{b}{1 + n_j b}(a+n_j\lambda_j ) .\label{eq : e_lambdaeb_lim_n}\ ] ] to second order and \longrightarrow \left ( \frac{b}{1+n_jb }  \right)^2 n_j\lambda_j , \label{eq : var_lambdaeb_lim_n}\ ] ] to first order .   the closed - form update of the ad - hoc distributed estimator allows us to perform a more detailed analysis . in particular, we are able to characterize mean and variance of the local estimator during the algorithm evolution ( transient analysis ) and for any fixed value of the number of nodes .  as we have done for the empirical bayes distributed estimator , also for the ad - hoc one we first characterize the estimator of the hyperparameter .before addressing the transient analysis , we compute mean and variance of the consensus value .the homogeneous estimator of , , is unbiased , i.e. , =b,\ ] ] and has variance = \frac{b}{a n } + \frac{b^2}{a n^2 } \sum_{i=1}^{n } n_i^2 \ , .\label{eq : var_bhom}\ ] ] moreover , the estimator is consistent , i.e. , it converges in probability to the true value as .[ prop : var_bhom ]  the proof is given in appendix [ sec : prop_var_bhom ] .the estimator is the ml estimator in the homogenous case , i.e. , when for all , and it attains the cramer - rao bound ( crb ) not only asymptotically , but for any .it follows by substituting in and .interestingly , we will show in the numerical analysis that even in the non - homogeneous scenario the ml estimator approaches the crb at any fixed .the following lemma gives a characterization of the transient local estimates .[ lem : e_var_bhom_t ] let assumption [ assum : graph ] and assumption [ assum : col_stochastic ] hold . for all ,the local estimator used in the ad - hoc distributed estimator ( algorithm [ alg : ad - hoc_estimator ] ) is unbiased at any , i.e. , =b,\ ] ] and has variance = \frac{b}{a } \frac{\sum_{k=1}^n \phi_{ik}(t)^2       n_k}{{\left(\sum_{k=1}^n \phi_{ik}(t ) n_k\right)^2 } } + \frac{b^2}{a }      \frac { \sum_{k=1}^n \phi_{ik}(t)^2 n_k^2}{\left(\sum_{k=1}^n        \phi_{ik}(t ) n_k\right)^2 }     \end{split}\ ] ] where is the element of the state transition matrix defined in .moreover , as , the variance of converges exponentially to the variance of , , and satisfies -{\mathrm{var}}[{\hat{b}^{\text{hom}}}]| \leq \frac{b(1 + 2b{n_{\text{max}}})}{\mu a }     \delta(\phi(t ) ) ,     \label{eq : bound_var_bhom_t}\ ] ] where is given in lemma [ lem : weak_ergodicity ] and is a ( proper ) coefficient of ergodicity continuous on the set of raw ( respectively column ) stochastic matrices satisfying .it is proper if if and only if , with ^t ] to parameters of the communication graph and the diffusion protocol as and .for example , if is balanced , the matrix is doubly stochastic , hence .  with this characterization of the hyperparameter estimator , we are ready to analyze the estimator . recalling the assumption that the measurements of agent do not contribute to the computation of , we clearly have uses as the estimate of some neighbor , participating to the distributed computation . ] = { \mathrm{e}}[{\hat{b}^{\text{hom}}}_j(t ) ] = b.\end{aligned}\ ] ] and =  { \mathrm{var}}[{\hat{b}^{\text{hom}}}_j(t ) ] .\ ] ]  it is worth noticing once more that although for a tractable , rigorous analysis we need the assumption that agent does not contribute to the distributed computation , in practice the results hold with good approximation also in the scenario in which agent contributes to the distributed computation .this is due to the weak impact of single measurements onto the aggregate quantities .in fact , for example , following analogous calculations as in the proof of lemma [ lem : e_var_bhom_t ] , the conditional mean in this latter case turns out to be = b - \frac{\phi_{jj}(t ) n_j}{\eta_j(t ) } \left(b    - \frac{\lambda_j}{a}\right),\ ] ] where is , again , the element of and we have also used that , conditioned to the of agent , = \left\ { \begin{array}{ll } \lambda_j & i = j\\ ab    & i\neq j\end{array } \right . .\ ] ] since as , , then the difference between equations and is practically negligible .  in the next theoremwe provide explicit transient expressions for the conditional mean and variance of .again , we use second - order and first - order approximations respectively .  [thm : mean_var_ad - hoc_estimator ] consider a network of monitors as in section [ sec : prob_form ] running the ad - hoc distributed estimator ( algorithm [ alg : ad - hoc_estimator ] ) . then the transient conditional mean and variance of are given by \approx \nonumber\\ & ( a+n_j\lambda_j)\left [ \frac{b}{1+n_j b }   - \frac{n_j}{(1+n_j b)^3 } { \mathrm{var}}[{\hat{b}^{\text{hom}}}_j(t ) ]  \right ] \label{eq : e_cond_approx_ah}\end{aligned}\ ] ] and \approx \nonumber\\  & \left [ \frac{b}{1+n_jb } -  \!\frac{n_j}{(1+n_jb)^3 } { \mathrm{var}}[{\hat{b}^{\text{hom}}}_j(t ) ] \right]^2 n_j\lambda_j \nonumber\\  & \;\ ; + \left[\frac{2n_j}{(1+n_jb)^3}\right]^2 { \mathrm{var}}[{\hat{b}^{\text{hom}}}_j(t)]\left [   ( a+n_j\lambda_j)^2+n_j\lambda_j\right]\label{eq : var_cond_approx_ah}\end{aligned}\ ] ] where indicates respectively the second - order and the first - order approximations , and  ] .in this section we analyze the performance of the proposed estimators . starting from the theoretical characterization developed in the previous section ,we perform a monte carlo analysis confirming the theoretical bounds and adding other insights on the performance of the estimators .  as performance metricwe adopt the root mean square error ( rmse ) , thus taking into account both bias and variance of the estimators .we recall that for an estimator of a parameter , the rmse is defined as & = \sqrt{{\mathrm{e}}[(\hat\omega - \omega)^2 ] } \nonumber\\  & = \sqrt{{\mathrm{var}}[\hat\omega ] + { \mathrm{e}}^2[\hat\omega ] - 2\omega{\mathrm{e}}[\hat\omega ] +   \omega^2 } .\end{aligned}\ ] ] clearly , if the estimator is unbiased the rmse coincides with the standard deviation , i.e. , = \sqrt{{\mathrm{var}}[\hat\omega]} ] , will be compared with the sample value obtained through the monte carlo trials , , computed as - \omega)^2},\ ] ] where is the number of trials and  ] with probability .  in order to challenge the ad - hoc distributed estimator we focus on a strongly inhomogeneous network scenario .that is , we consider a network in which half of the nodes have the maximum number of measurements in the observation window , ( we set ) , and the remaining ones only one measurement , .we start by analyzing the transient performance of the ad - hoc distributed estimator .consistently with the theoretical analysis , we first focus on the time evolution of the rmse of .notice that , since is an unbiased estimator it holds = \sqrt{{\mathrm{var}}[{\hat{b}^{\text{hom}}}_i(t)]} ] for nodes , two of them with measurements and two with one measurement .the theoretical curve predicts very accurately the sample rmse obtained by the monte carlo trials , as highlighted in the inset .as expected , the rmse of the different nodes converges to the consensus value  ] with their sample counterparts .we can highlight two main differences with respect to the previous scenario .the curves have some constant portions showing that nodes can be isolated for some time - intervals. however , the convergence is faster compared to the fixed scenario .this can be explained by the higher density of the union graph in the time - varying scenario as opposed to the sparsity of the fixed graph .in fact , we noticed that increasing the erds - rnyi graph parameter increases the convergence speed . ] under time - varying communication graph .the solid lines indicate the theoretical expressions , while the dash - dot lines are the ones obtained via monte carlo simulations .the dashed horizontal line is the theoretical consensus value.,scaledwidth=40.0% ]   ] for a generic node .notice that we use the index , rather than , meaning that in the monte carlo trials the node under investigation participates to the computation of .this allows us to show that the uncorrelation assumption in section [ sec : theoretical_analysis ] is in fact reasonable .moreover , the computations will also confirm the validity of the low - order approximation made to derive the theoretical expressions of mean and variance .  in figure[ fig : lambda_transient_fixed ] we compare the theoretical and sample evolution of  ] , decreases , since the hyperparameter is estimated by means of a larger sample .this aspect will be better highlighted in the following asymptotic analysis in which we focus on how  ] practically coincide with the sample ones .although for the ml estimator we have no theoretical expression for fixed , the picture shows a very interesting property .that is , the ml estimator achieves the crb not only asymptotically ( ) as predicted by the theory , but also for each fixed .interestingly , in accordance to the theoretical results in the previous section , also the homogeneous estimator achieves the crb as goes to infinity .finally , we analyze the rmse of the estimators of .we consider an agent with one measurement ( ) and use the most frequent value for the arrival rate , i.e. , the mode of the gamma distribution , .  in figure[ fig : lambda_asympt_n ] we plot the sample rmse of the empirical bayes and ad - hoc estimators .we also plot the ( sample ) values of the decentralized estimator .we decided to normalize all the curves to the theoretical value of the decentralized estimator in order to highlight the improvements of the proposed distributed estimators .clearly , the sample values of the decentralized estimator are approximately equal to one , with minor fluctuations only due to the finite number of samples .we compare the sample curves with the theoretical curve of the homogeneous estimator and with the theoretical asymptotic value as .the theoretical curve is obtained as follows :  ] can be computed by plugging  ] is obtained by plugging  ] into .  again , although computed under the uncorrelation assumption and neglecting higher order terms , the theoretical expression  ] , so that , after some manipulation \\  & = \frac{a}{b^2}\sum_{i=1}^n \frac{n_ib}{n_ib+1},\end{aligned}\ ] ] so that the proof follows .the mmse estimator of in can be written as , where and . due to the independence between and , we have ={\mathrm{e}}[zy |\lambda_j]={\mathrm{e}}[z |\lambda_j ] { \mathrm{e}}[y  |\lambda_j ]  \label{eq : tot_e_cond_ml}\ ] ] and &={\mathrm{e}}^2[z |\lambda_j ] { \mathrm{var}}[y |\lambda_j]\nonumber\\  & \quad+ \!{\mathrm{var}}[z |\lambda_j ] ( { \mathrm{e}}^2[y |\lambda_j ] + { \mathrm{var}}[y  |\lambda_j ] ) .\label{eq : tot_var_cond_ml}\end{aligned}\ ] ] the conditional moments of are easily obtained =a+n_j \lambda_j  \label{eq : e_y}\ ] ] and =n_j\lambda_j .\label{eq : var_y}\ ] ] for the nonlinear function we resort to an approximate analysis .that is , we consider the taylor expansion for the moments of the function around the mean value = { \mathrm{e}}[{\hat{b}^{\text{ml}}}] ] and \rightarrow 0 ] , thus & = { \mathrm{e}}[{\mathrm{e}}[\sigma | \bm{\lambda } ] ] = { \mathrm{e}}\left [ \sum_{i=1}^n { \mathrm{e}}[\sigma_i |   \lambda_i ] \right]= { \mathrm{e}}\left[\sum_{i=1}^n n_i\lambda_i\right]\\  & = \sum_{i=1}^n n_i a b = n a b,\end{aligned}\ ] ] so that =b ] , so that equation follows . by markov s inequality and lemma [ prop : var_bhom ] ,\ ] ] and the variance can be bounded as follows & = \frac{b}{a n } +  \frac{b^2}{a n^2 } \sum_{i=1}^{n } n_i^2\\  & \leq \frac{b}{a n } + \frac{b^2}{a n^2 }  \sum_{i=1}^{n } { n_{\text{max}}}^2\\ & = \frac{b}{a n } + \frac { { { n_{\text{max}}}}^2 b^2}{a n } \ ;  \stackrel{n \rightarrow \infty}{\xrightarrow{\hspace*{1 cm } } } \ ; 0 \end{split}\ ] ] so that in probability , thus concluding the proof .   to prove that is unbiased at any ,first let us recall that the aggregate states ^t ] evolve according to the dynamics with ^t { \stackrel{\textrm{\tiny def}}{=}}{\bm{\sigma}} ] and the column stochastic matrix with elements .the evolution of and is given by where is the deterministic state transition matrix defined in .next , observe that the update depends only on the initial value and thus is deterministic .therefore , using the update in algorithm [ alg : ad - hoc_estimator ] , we have = \frac{1}{a \eta_i(t ) } { \mathrm{e}}[s_i(t)]\ ] ] using the evolution of the aggregate state and denoting the canonical vector ( e.g. , ^t ] it follows & = \frac{1}{a \eta_i(t ) } e_i^t \phi(t ) a b \ , \bm{n}\\            & = \frac{b}{\eta_i(t ) } e_i^t \eta(t ) = b\\ \end{split}\ ] ] where the last two steps follow respectively from ( ) and from .   using again the update in algorithm [ alg : ad - hoc_estimator ] ,it holds & =  \frac{1}{a^2 \eta_i(t)^2 } { \mathrm{var}}[e_i^t ( \phi(t){\bm{\sigma}})].\\ \end{split}\ ] ] noting that is the row of , it follows & =  \frac{1}{a^2 \eta_i(t)^2 } { \mathrm{var}}\left[\sum_{k=1}^n \phi_{ik}(t ) \sigma_k\right ] , \end{split}\ ] ] where is the element of the matrix . by the law of total variance we have that & =  \frac{1}{a^2 \eta_i(t)^2 } \left\ { { \mathrm{e}}\left[{\mathrm{var}}\left[\sum_{k=1}^n \phi_{ik}(t )    \sigma_k \;\big |\ ; { \bm{\lambda}}\right]\right ] \right.\\  & \qquad\left .+ { \mathrm{var}}\left [ { \mathrm{e}}\left[\sum_{k=1}^n \phi_{ik}(t ) \sigma_k  \;\big |\ ;    { \bm{\lambda}}\right ] \right ] \right\}\\  & = \frac{1}{a^2 \eta_i(t)^2 } \left\ {  { \mathrm{e}}\left[\sum_{k=1}^n \phi_{ik}(t)^2 n_k    \lambda_k \right]\right.\\    & \left.\qquad +  { \mathrm{var}}\left[\sum_{k=1}^n \phi_{ik}(t ) n_k    \lambda_k \right ] \right\ } , \end{split}\ ] ] where we have used = n_k \lambda_k ] . finally , recalling that = a b ] and the are independent , it turns out & = \frac{b}{a \eta_i(t)^2 } \left ( \sum_{k=1}^n \phi_{ik}(t)^2n_k + b \sum_{k=1}^n \phi_{ik}(t)^2 n_k^2 \right ) , \end{split}\ ] ] so that equation  follows .  to prove the asymptotic result , we work out - { \mathrm{var}}[{\hat{b}^{\text{hom}}}]| ] follows .then , from the definition of the coefficient of ergodicity , it follows |  \nonumber \\&\leq\frac{1}{\sum_{k=1}^n \phi_{ik}(t ) n_k } \left ( \frac{b}{a n }   n  \delta(\phi(t ) ) + \frac{b^2}{a n^2 }   n^2 { n_{\text{max}}}2 \delta(\phi(t ) ) \right)\end{aligned}\ ] ] where we have simplified the common factor . finally , by lemma [ lem : weak_ergodicity ] , , with so that |  & \leq  \frac{\delta(\phi(t ) ) b(1 + 2b{n_{\text{max}}})}{\mu a } \ ] ] thus concluding the proof .   the local update of the ad - hoc distributed estimator ( algorithm [ alg : ad - hoc_estimator ] ) can be written as , where and . using again the independence between and ,we can write  ] as = { \mathrm{e}}[xy |\lambda_j ] = { \mathrm{e}}[x |\lambda_j ] { \mathrm{e}}[y |\lambda_j ]    \label{eq : tot_e_cond_ah}\ ] ] and ={\mathrm{e}}^2[x |\lambda_j ] { \mathrm{var}}[y |\lambda_j]\nonumber\\                   & + { \mathrm{var}}[x |\lambda_j ] ( { \mathrm{e}}^2[y |\lambda_j ] \!+\!{\mathrm{var}}[y |\lambda_j ] ) .\label{eq : tot_var_cond_ah }  \end{aligned}\ ] ] considering , as in the previous theorem , the taylor expansion for the moments of the function around the mean =b ] is the one given in and , again , the symbol indicates that we have neglected higher - order terms in the expansion of .hence , plugging and in , equation follows .similarly , \approx \left(x'({\mathrm{e}}[{\hat{b}^{\text{hom}}}_j(t)])\right)^2  { \mathrm{var}}[{\hat{b}^{\text{hom}}}_j(t ) ] \nonumber\\  & = \left [ \frac{1}{(1+n_j b)^2 } \right]^2  \left ( \frac{b}{a n } +  \frac{b^2}{a n^2 } \sum_{i=1}^{n } n_i^2\right ) .\label{eq : var_x}\end{aligned}\ ] ] using - into , equation follows , thus concluding the first part of the proof .a. coluccia and g. notarstefano , `` distributed bayesian estimation of arrival rates in asynchronous monitoring networks , '' in _ ieee international conference on acoustics , speech and signal processing ( icassp ) _ , 2014 , pp .50505054 .s. barbarossa , s. sardellitti , and p. di lorenzo , _ distributed detection and estimation in wireless sensor networks _ , ser .communications and radar signal processing.1em plus 0.5em minus 0.4emacademic press library in signal processing , october 2013 , vol . 2 , ch . 7 , pp . 329408 .i. d. schizas , a. ribeiro , and g. b. giannakis , `` consensus in ad hoc wsns with noisy links part i : distributed estimation of deterministic signals , '' _ ieee transactions on signal processing _ ,56 , no . 1 ,pp . 350364 , 2008 .p. frasca , h. ishii , c. ravazzi , and r. tempo , `` distributed randomized algorithms for opinion formation , centrality computation and power systems estimation : a tutorial overview , '' _european journal of control _, 2015 .s. barbarossa and g. scutari , `` decentralized maximum - likelihood estimation for sensor networks composed of nonlinearly coupled dynamical systems , '' _ ieee transactions on signal processing _ , vol .55 , no . 7 , 2007 .a. chiuso , f. fagnani , l. schenato , and s. zampieri , `` gossip algorithms for simultaneous distributed estimation and classification in sensor networks , '' _ ieee journal of selected topics in signal processing _ ,vol . 5 , no . 4 , pp .691706 , 2011 .f. fagnani , s. m. fosson , and c. ravazzi , `` input driven consensus algorithm for distributed estimation and classification in sensor networks , '' in _ 50th ieee conference on decision and control and european control conference ( cdc - ecc)_.1em plus 0.5em minus 0.4emieee , 2011 , pp .66546659 .d. varagnolo , g. pillonetto , and l. schenato , `` distributed consensus - based bayesian estimation : sufficient conditions for performance characterization , '' in _ american control conference_.1em plus 0.5em minus 0.4emieee , 2010 , pp .39863991 .  p. dilorenzo and s. barbarossa , `` distributed least mean squares strategies for sparsity - aware estimation over gaussian markov random fields , '' in _ ieee international conference on acoustics , speech and signal processing ( icassp ) _ , 2014 , pp .54725476 .g. mateos , i. d. schizas , and g. b. giannakis , `` distributed recursive least - squares for consensus - based in - network adaptive estimation , '' _ ieee transactions on signal processing _ , vol .57 , no . 11 , pp . 45834588 , 2009 .i. d. schizas , g. b. giannakis , s. i. roumeliotis , and a. ribeiro , `` consensus in ad hoc wsns with noisy links part ii : distributed estimation and smoothing of random signals , '' _ ieee transactions on signal processing _ ,56 , no . 4 , pp . 16501666 , 2008 .f. zanella , d. varagnolo , a. cenedese , g. pillonetto , and l. schenato , `` asynchronous newton - raphson consensus for distributed convex optimization , '' in _3rd ifac workshop on distributed estimation and control in networked systems ( necsys12 ) _ , 2012 .j. n. tsitsiklis , d. p. bertsekas , m. athans _et al . _ ,`` distributed asynchronous deterministic and stochastic gradient optimization algorithms , '' _ ieee transactions on automatic control _ , vol . 31 , no . 9 , pp . 803812 , 1986 .a. jadbabaie , j. lin , and a. s. morse , `` coordination of groups of mobile autonomous agents using nearest neighbor rules , '' _ ieee transactions on automatic control _ ,48 , no . 6 , pp .9881001 , 2003 .f. bnzit , v. blondel , p. thiran , j. tsitsiklis , and m. vetterli , `` weighted gossip : distributed averaging using non - doubly stochastic matrices , '' in _ ieee international symposium on information theory proceedings ( isit)_.1em plus 0.5em minus 0.4emieee , 2010 , pp .17531757 .n. h. vaidya , c. n. hadjicostis , and a. d. dominguez - garcia , `` distributed algorithms for consensus and coordination in the presence of packet - dropping communication links - part ii : coefficients of ergodicity analysis approach , '' _ arxiv preprint arxiv:1109.6392 _ , 2011 .[ ] angelo coluccia ( m13 ) received the eng .degree in telecommunication engineering ( summa cum laude ) in 2007 and the phd degree in information engineering in 2011 , both from the university of salento , lecce , italy .former researcher at forschungszentrum telekommunikation wien , vienna , since 2008 he has been engaged in research projects on traffic analysis , security and anomaly detection in operational cellular networks .he is currently assistant professor at the dipartimento di ingegneria dellinnovazione , university of salento , where he teaches the course of telecommunication systems .his research interests are signal processing , communications and wireless networks , in particular cooperative sensing / estimation approaches for localization and other ( possibly distributed ) applications .[ ] giuseppe notarstefano has been an assistant professor ( ricercatore ) at the universit del salento ( lecce , italy ) since february 2007 .he received the laurea degree `` summa cum laude '' in electronics engineering from the universit di pisa in 2003 and the ph.d .degree in automation and operation research from the universit di padova in april 2007 .he has been visiting scholar at the universities of stuttgart , california santa barbara and colorado boulder .his research interests include distributed optimization , cooperative control in multi - agent networks , applied nonlinear optimal control , and trajectory optimization and maneuvering of aerial and car vehicles .he serves as an associate editor in the conference editorial board of the ieee control systems society and for the european control conference , ifac world congress and ieee multi - conference on systems and control .he coordinated the vi - rtus team winning the international student competition virtual formula 2012 .he is recipient of an erc starting grant 2014 .

Abstract: in this paper we consider a network of agents monitoring a spatially distributed arrival process . each node measures the number of arrivals seen at its monitoring point in a given time - interval with the objective of estimating the unknown local arrival rate . we propose an asynchronous distributed approach based on a bayesian model with unknown hyperparameter , where each node computes the minimum mean square error ( mmse ) estimator of its local arrival rate in a distributed way . as a result , the estimation at each node `` optimally '' fuses the information from the whole network through a distributed optimization algorithm . moreover , we propose an _ ad - hoc _ distributed estimator , based on a consensus algorithm for time - varying and directed graphs , which exhibits reduced complexity and exponential convergence . we analyze the performance of the proposed distributed estimators , showing that they : ( i ) are reliable even in presence of limited local data , and ( ii ) improve the estimation accuracy compared to the purely decentralized setup . finally , we provide a statistical characterization of the proposed estimators . in particular , for the ad - hoc estimator , we show that as the number of nodes goes to infinity its mean square error converges to the optimal one . numerical monte carlo simulations confirm the theoretical characterization and highlight the appealing performances of the estimators .   distributed estimation , empirical bayes , push - sum consensus , cyber - physical systems .


in recent years , smart grid has attracted significant attention in the field of power systems , communications and networking . in a smart grid ,power is delivered from power suppliers to home appliances with the aid of two - way communications .the price of power changes with time , subject to many random factors like congestion level and power generation .home appliances can inquire the instantaneous price and decide the consumption level of power .for example , at midnight , the power load is usually low and thus the price is relatively low ; therefore , air conditioner may set a higher temperature ( suppose that it is in the winter ) and enjoy the low power price .existing studies usually pay attention to only the cost of power consumption and ignore the cost of communications .however , communications for inquiring the price in smart grid could incur a nonnegligible cost .therefore , it is necessary to consider the cost of communications and study the optimal policy of requesting communications for power price inquiry , such that the total cost of power consumption and communications is minimized under certain constraints . to the authors best knowledge, no existing work has incorporated the cost of communication into the decision procedure in smart grid .  due to the cost of communication, it may not be optimal to inquire the power price frequently .if the power price changes slowly , it may be better to use old power price to optimize the power consumption level .however , using old power price is also risky .for example , when the old power price is much higher than the current power price , the home appliance may use a lower power consumption , thus wasting the opportunity of low power price if it still uses the old price and does not inquire the new one .when the old power price is much lower than the current price , the policy of using old price will result in a high power consumption level , thus incurring significant penalty .therefore , an optimal tradeoff should be found between the cost of communications and the penalty incurred by using the old value of power price .  in this paper, we study the decision problem of communication for inquiring the power price in order to minimize the total cost . a key issue is the prediction of power price based on the current obtained price .a model called _ probabilistic locational marginal pricing ( lmp ) forecasting _ is applied to predict the distribution of the power price at a given future time . in this model ,a curve is used to map from the true value of power load to the power price .a probabilistic model similar to brownian motion is used to describe the distribution of load as a functional of the elapsed time since the latest inquiry of the power price .once the evolution law of the power price ( load ) is known , we convert the problem into a markov decision process ( mdp ) and obtain the corresponding state transition probabilities . then, we apply dynamic programming to compute the optimal strategy . to simplify the strategy, we also propose a myopic strategy which compares the cost of communication with the penalty incurred by using the old power price .note that we assume that the dynamics of the power price are perfectly known .for practical case , when there is no perfect model for the power price , we can apply the approach of reinforcement learning to learn the optimal strategy , which is beyond the scope of this paper .the remainder of the paper is organized as follows . the system model and the price forecastingare introduced in section [ sec : system ] .the optimal communication policy is discussed in section [ sec : inquiry ] .numerical results are provided in section [ sec : numerical ] while conclusions are drawn in section [ sec : conclusion ] .in this section , we first introduce the system model used in this paper . then, we give a brief introduction to the pricing mechanism in the power grid , namely the lmp forecasting mechanism .practical power grid is very complicated . to simplify the analysis andfacilitate the analytical discussion , we consider a simple supplier - home model , as illustrated in fig .[ fig : model ] . in this model, we consider only one power supplier and one home appliance .the power supplier provides power for the home appliance via power line , as well as the power price via a communication channel .the communication channel could be over internet , wireless networks or power line communication systems .the home appliance inquires the power price via the communication channel .the following assumptions are used throughout the paper :  * we ignore possible errors over the communication channel and assume perfect communications .we do not consider the details of communications like modulation and coding .* time is divided into time slots . at the beginning of each time slot, the power supplier adjusts its power price .then , the home appliance can inquire the power price and receive the price information without or with delay .once the price is obtained , the home appliance determines its power consumption level .these events are illustrated in fig .[ fig : timing ] . * for simplicity , we assume that the appliance can adjust its power consumption according to the power price immediately . in practice , there could be a delay for the power consumption adjustment , e.g. it needs some time to start the air conditioner .the corresponding analysis will be more complicated and is beyond the scope of this paper .we denote by the power price and the power consumption at the -th time slot .the utility function of the power consumption is denoted by , i.e. the home receives reward when the power consumption is . for simplicity , we assume that the utility function does not change with time .the cost for one communication effort is denoted by , which is assumed to be a constant .we denote by the event that the home inquires the power price at the -th time slot , i.e. if it inquires and otherwise .we denote by the latest time slot before time slot in which the home inquired the power price .the home appliance uses the same power price since the previous price inquiry , namely at time slot .therefore , the decision of power consumption is based on the power price of the previous price inquiry , i.e. is a function of . for simplicity , we assume that the power consumption level maximizes the net reward and ignores the communication cost , i.e. where is the price assumed by the home appliance ( may be different from the true value if the home appliance does not inquire the power price ) .we assume that is an increasing and strictly concave function ( thus the first order derivative is strictly decreasing ) .we also assume that is continuously differentiable and its first order derivative , denoted by ranges from to .therefore , the optimal value of the power consumption level is given by which is derived from the first order condition , i.e. since ranges from to 0 and is continuous and strictly decreasing , there exists a unique solution to ( [ eq : price ] ) .hence , we have a one - to - one mapping between the price and the optimal power consumption level .although there are some simplifications in these quantities , we can obtain the insight from the simplified model and extend them to more generous case in the future .based on the above definitions , we assume that the total reward of the home appliance is the discounted sum of the rewards in different time slots , which is given by where is the discount factor .     power price is usually determined by lmp methodology , which is actually driven by the time - varying load , as illustrated in fig .[ fig : factors ] .the mapping between load and lmp is typically obtained from a constrained optimization problem . in practice, the mapping can be represented by a piecewise curve , as illustrated in fig .[ fig : mapping ] .therefore , the uncertainty of the power price is from that of load .we notice that , in fig .[ fig : mapping ] , the number of possible prices is finite. therefore , we denote by the total number of possible prices and by , , ... , the corresponding prices .meanwhile , we denote by , ... , the load intervals corresponding to the prices . given a price , we assume that the load is uniformly distributed within the interval .    actually the load is a random variable .typically , it is modeled as a gaussian random variable , i.e. where is the load at time slot , and are the corresponding expectation and variance .note that the assumption of gaussian random variable is an approximation since a gaussian random variable could be negative while a negative price is nonsense . to make it mathematically rigorous, we modify the probability density function ( pdf ) of as where is the maximal possible load .  to study the optimal inquiry time for communications , we need to model the relationship between the time interval and the gaussian distribution parameters .suppose that , at time slot 0 , the true value of the load , , is known and then there is no observation on the true value of the load. then , we use the following assumptions for the load distribution at time slot :  * the expectation equals , i.e. the expectation does not change with time , which represents an unbiased price prediction .* the variance equals , i.e. the variance increases linearly with the time gap , similarly to a brownian motion .the parameter can be estimated from historical data .the rationale behind the brownian motion like variance is that brownian motion is widely used to drive the price fluctuation , e.g. stochastic differential equation , in the area of financial analysis .therefore , we also use a linearly increasing variance to model the increasing uncertainty with time , which stimulates the price inquiry over the communication channel .in this section , we study the optimal policy for price inquiry over the communication channel .we can model the decisions of price inquiry as a markov decision process ( mdp ) .we will discuss the fundamental elements in the mdp .then , we apply the approach of value iteration to obtain the optimal policy .note that we assume that the statistical laws of the price , like the mapping between the load and the price and the uncertainty on the load , are all known to the home appliance .there are three elements in a mdp problem , namely action space , system state and reward .we discuss them in the context of price inquiry separately .   obviously , the action of the home appliance is the inquiry of the power price ( denoted by 1 ) or not ( denoted by 0 ) .the decision of action is determined by the current system state and the policy of the home appliance .actually , there is an implicit action for the home appliance , i.e. the power consumption level . in the most complicated case, the power consumption level should also be a function of the current system state and policy .however , to simplify the analysis , we assume that the home appliance assumes the power price of the latest inquiry , thus uniquely determines the power consumption level .therefore , we do not consider the power consumption level as an action and do not incorporate it into the decision policy .we put an upper bound , denoted by , for the number of time slots between two price inquiries .then , the home appliance must send out a price inquiry within time slots since the previous price inquiry .the upper bound can address the jeopardy brought by modeling imperfection , e.g. some imprecise parameters may significantly lengthen the interval between two inquiries to a harmful level , and thus improves the robustness .the system state contains two parts , namely the power price in the previous inquiry and the elapsed time since the previous inquiry . for timeslot , the system state is given by .obviously , if the home appliance inquires the power price at time slot , the corresponding system state is .an illustration for the state transition diagram is shown in fig .[ fig : state ] for the case of two possible power prices ( high or low ) .we notice that , whenever the price substate is changed , the substate of elapsed time is reset to 0.    another important issue in the system state is the state transition probability respect to the action . obviously , if the action is no inquiry , i.e. 0 , the state is changed to if the previous state is .when the action is inquiry , i.e. 1 , the state is changed to from the previous state , where is a random variable .we denote by the probability of transiting from price to price after time slots .the transition probability is then given by where is the length of interval and is the conditional pdf of given . from ( [ eq : evolution ] ) and the assumption on the expectation and variance , it is easy to verify that the conditional pdf is given by    suppose that the system state of the previous time slot is . if the action of the current time slot is 0 , i.e. no inquiry , the expected reward of the current time slot is given by ( recall that the power consumption level is a function of the assumed price ) otherwise , the reward is given by ( recalled that is the cost of communication )    once defining the elements of mdp , we can apply dynamic programming ( dp ) to obtain the optimal strategy .we denote by the optimal expected total reward when the initial state is .then , satisfies the following bellman s equation , which is given by \right),\end{aligned}\ ] ] where is the expected reward due to action and system state , is the expectation conditioned on and , is the system state in the next time slot . the expectation can be computed using the state transition probability in ( [ eq : transition ] ) .the instantaneous reward can be computed using ( [ eq : reward1 ] ) and ( [ eq : reward2 ] ) .the bellman s equation can be solved by using the following value iteration , which is given by \right),\end{aligned}\ ] ] where the superscript is the index of iteration .the iteration converges to the solution of the bellman s equation as .the optimal action is obtained from \right).\end{aligned}\ ] ]   in the bellman s equation ( [ eq : bellman ] ) , the optimal action based on the current system state should take the future reward into account , thus making the solution complicated . a simpler but perhaps suboptimal scheme is the myopic strategy , i.e. optimizing the instantaneous reward without considering the future system state evolution .then , the corresponding action is given by i.e. do not inquire the power price if the reward of no inquiry is larger than that of inquiry , and vice versa . by applying the expressions of rewards in ( [ eq : reward1 ] ) and ( [ eq : reward2 ] ) , we have where is defined as which stands for the penalty of using the old value of the power price .then , the decision rule ( [ eq : myopic ] ) means that , when the cost of communication is larger than the penalty of using the old power price instead of the new one , the action should be no price inquiry ; and the home appliance should inquire the power price , otherwise .in this section , we use numerical simulations to explore the optimal power price inquiry based on the above framework .we use the pjm five - bus power system for simulations , whose configuration is illustrated in fig . [fig : pjm ] . the corresponding curves of lmp versus load for the five busesare shown in fig .[ fig : curve ] and the lower boundaries for the load intervals are given in table [ tab : curve ] ( note that ) .both the curves and the data are obtained from the continuous lmp model in .      .lmp( $ /mwh ) versus load ( mw ) [ cols="^,^,^,^,^,^",options="header " , ]   we assume that the upper bound of time slots between two inquiries is 10 time slots .we set for the discounted sum of cost as the performance metric .the utility function of the home appliance is assumed to be .the default values of and are set to 200 and 10 .we first use the value iteration to obtain the optimal strategy and the corresponding minimum discounted sum of cost .note that the cost here is defined as the gap of the corresponding reward to the ideal reward when there is no communication cost .we also tested the performance of the strategy of inquiring the power price in every time slot .the ratios of the discounted sum of cost obtained from the optimal strategy and that of the always - inquiry strategy are then computed for different values of and are shown in fig .[ fig : cost_ratio ] .there are five curves since there are five buses in the simulated power system .obviously , the smaller the ratio is , the better the performance gain of the optimal strategy of price inquiry is .we observe that , for buses a , b and c , the ratio ranges between 0.6 and 0.8 , i.e. the optimization can decrease the total cost by 20% to 40% .the performance gain of bus d is smaller .the reason is that the price changes the most radically for bus d , thus requiring more price inquiries .since the power price of bus e has only marginal changes with respect to the change of load , it is much less necessary to inquire the price , thus making the performance gain of bus e much more significant than other buses .  .]  then , we compare the performance of the optimal strategy with that of the no - inquiry strategy .the curves of cost ratios are shown in fig .[ fig : cost_ratio2 ] . in sharp contrast to fig .[ fig : cost_ratio ] , the order of the performance gain is reversed .the performance gain of bus e is the least since the necessity of price inquiry is the least for bus e. however , even for bus e , where the price changes only marginally , the optimal strategy can still achieve a very significant gain , thus demonstrating the necessity of price inquiry in smart grid .  .]  we repeated the simulations in figures [ fig : cost_ratio ] and [ fig : cost_ratio2 ] for different communication costs and fixed . from both figures ,the impacts of increasing communication cost on the performance gain are contrary . as the communication cost increases, the home appliance should be more inclined not to inquire the power price to reduce the cost of communication .therefore , the performance gain in fig .[ fig : cost_ratio3 ] increases while that in fig .[ fig : cost_ratio4 ] is decreased .      .]     in figures [ fig : cost_ratio5 ] and [ fig : cost_ratio6 ] , we compare the performance of the optimal strategy and the myopic strategy with different or different communication costs . in the ratio of cost , the numerator is the cost of the optimal strategy while the denominator is that of the myopic strategy .we observe that the myopic strategy is very close to optimal .therefore , in practical systems , one may consider the myopic strategy due to its simplicity .we have considered the communication cost in smart grid , which is often omitted in existing studies .the dynamics of power price have been modeled as a markov chain by modeling the random process of load as a brownian motion like one and employing the lmp - load mapping curve .then , the decision of power inquiry has been considered as a mdp problem and dynamic programming is employed to compute the optimal strategy . to avoid the high computational cost ,we have studied a simple and suboptimal myopic strategy .a pjm five - bus system has been used for numerical simulation , which shows significant performance gain of the optimal strategy of price inquiry , as well as the near - optimality of the myopic approach .

Abstract: in smart grid , a home appliance can adjust its power consumption level according to the realtime power price obtained from communication channels . most studies on smart grid do not consider the cost of communications which can not be ignored in many situations . therefore , the total cost in smart grid should be jointly optimized with the communication cost . in this paper , a probabilistic mechanism of locational margin price ( lmp ) is applied and a model for the stochastic evolution of the underlying load which determines the power price is proposed . based on this framework of power price , the problem of determining when to inquire the power price is formulated as a markov decision process and the corresponding elements , namely the action space , system state and reward function , are defined . dynamic programming is then applied to obtain the optimal strategy . a simpler myopic approach is proposed by comparing the cost of communications and the penalty incurred by using the old value of power price . numerical results show the significant performance gain of the optimal strategy of price inquiry , as well as the near - optimality of the myopic approach .


an extensive literature is available on applications of stochastic orders to finance and insurance markets .the implications of stochastic orders for derivative pricing and risk management are relevant . the increasing dependence of european option prices by the riskiness of the underlying it is a well known property for basic models like that of black - merton - scholes , in which riskiness is expressed in terms of the logreturns distribution variance :the uncertainty is quantified there through the dispersion around the expected value and the distribution functions can be ordered according to their `` peakedness '' ; the larger is the dispersions , the higher the option prices .this very elementary and intuitive observation for simple models become more involved when turning attention to more complex models , where a more rigorous approach is necessary in order to avoid wrong conclusions .the black - merton - scholes model is nowadays considered fairly inadequate to describe the asset price dynamics ; several empirical facts can not be explained on the basis of this model : some statistical features exhibited by logreturns like fat tails , volatility clustering , aggregational gaussianity and the so - called leverage effect are completely outside of the prevision properties of the black - merton - scholes model . moreover a very relevant phenomenon exhibited by option prices , the `` volatility smile '' ( and its term structure ) can not be explained on this model basis . in order to provide a more satisfactory descriptionseveral different models have been introduced .some of these models introduce a stocastic dependence in volatility and/or jumps in asset logreturns ( and/or in volatility ) dynamics both in continuous and discrete time setting .  among discrete time models introduced in order to remove some of the black - merton - scholes model drawbacks ,the class of autoregressive conditioned heteroschedastic ( arch ) models introduced by engle and their general extension ( garch models ) proposed by bollerslev in bollerslev have risen considerable interest .several results related to stochastic orders are available for the continuous time models class : in , where a systematic investigation on semimartingale models is performed ; the models considered there include the heston and the barndorff - nielsen and shephard models . in . mller provides some results on stochastic orders in a dynamic reinsurance market where the traded risk process is driven by a compound poisson process and the claim amount is unbounded .stochastic order properties have been used to obtain bounds for option prices in incomplete markets ; the literature focused on this subject is quite extensive and we just mention the papers by el karoui et al . , by bellamy and jeanblanc , and by gushchin and mordecki .the purpose of the present paper is to present a systematic investigation of stochastic orders propagation in a garch context .comparison with stochastic orders in incomplete market models can give rise to different classes of problems : first can be considered the comparison of models under the same probability measure but with different parametric specification , second it can be examined the problem of comparing the same model under different probability measures ; as a matter of fact , when markets are incomplete , there are several probability measures equivalent to the historical one , under which the dynamics of prices can be given .in this paper we shall focus on the first class of problems mentioned : we shall provide a systematic comparison of logreturns and then of prices when the model parameters change , but the dynamics is specified under the same probability measure . in a garch context the parameters entering into play are three parameters assuming a numerical value and the innovations , which are random iid variables for which the density function is assigned .we just mentioned that stochastic order results have important implications on option pricing and this holds true also in a garch context : in particular , convex order relations on logreturn sums imply increasing convex order relations on the underlying price , hence on european call option prices and this can be considered the main relevance of our results from an application viewpoint .we like to present a numerical illustration in order to motivate our investigation .fig 1 compare the densities of the logreturn sums in a garch ( 1,1 ) model with respect to variations in the parameters ; in the continuous line all parameters assume the value 0.2 , while the other lines represent the same density , but with parameters , and respectively ( the parameter assuming the value 0.5 in indicated by a capital letter in each curve caption ) .the sums include the first 50 terms of the logreturn sequence .+  the innovations , assumed to be independent and identically distributed , and the initial variance , are chosen as standard gaussian random variables .it is evident how the usual real numbers order relation between parameters implies some ordering on logreturn sums ; it then arises as a natural question to ask if this simple remark can be made rigorous and if this conclusion can be cast into a more general framework considering different kind of stochastic orders and comparing stochastic order propagation from innovations to logreturns and to logreturns sums .  in the following section we present briefly the garch models and an auxiliary lemma . in section 3we present the univariate stochastic comparisons for logreturns in a garch setting . in section 4we provide results on some related orders , while in section 5 the convex order propagation of the logreturn sums is investigated together with its implications on asset and european call option prices , while their multivariate convex order propagation is the subject of section 6 .we focus our attention on the garch(1,1 ) case in the last section .the main results provided in this paper require the assumption of a symmetric probability density for innovations .the extension of the stochastic order propagation analysis presented here to the case of non - symmetric innovation densities would be of great interest : both the convex order propagation result and the comparison with respect to parameter variations will be the subject of future investigation in this more general setting . moreoverthe identification of some convex multivariate order which naturally propagate from innovations to logreturns is another target on which our research interest will be focused .these will be the subject of our future work .we consider garch models of two different very general forms ; the first model ( m1 ) is : =0 \\ \sigma _ { n+1}=f^{i}(\left\vert \varepsilon _ { n}\right\vert , \sigma _ { n})% \end{array}% \right .\label{garch1}\]]with increasing and componentwise convex ( ccx for brevity ) . the second model ( m2 ) is=0 \\ \sigma _ { n+1}^{2}=f^{ii}(\varepsilon _ { n}^{2},\sigma _ { n}^{2})% \end{array}% \right .\label{garch2}\]]with increasing and ccx . in both cases the innovations are independent and identically distributed ( i.i.d . ) .when discussing the propagation of variability orderings , the normalization requirement =1 ] for all increasing [ convex , increasing convex ] functions for which the expectations exist .we will see that in the general context of models m1 and m2 the orderings that are naturally propagated from the innovations to are the and the ordering _ between absolute values or squared variables_. this clearly completely modifies their interpretation ; in particular , in the next section we will see that the ordering between absolute values or squares can be interpreted as a _ variability ordering _ , while the ordering between absolute values or squares can be interpreted as a _kurtosis _ ordering .in order to establish these results , we proceed in two steps : first we consider the volatilities and then the variables .the first step is an immediate consequence of lemma [ giennel ] :  [ compsigma]*comparisons of * * * and * * ) let be as in ( [ garch1 ] ) and ; it follows that .b ) let be as in ( [ garch1 ] ) and ; it follows that .c ) let be as in ( [ garch2 ] ) and ; it follows that .d ) let be as in ( [ garch2 ] ) and ; it follows that .since from lemma [ giennel ] in model m1 with increasing and ccx , item a ) and b ) follows respectively fromtheorem 1.a.3 in and theorem 4.a.15 in .similarly , since from lemma [ giennel ] in model m2 with increasing and ccx , from the same theorems we get c ) and d ) .the comparison results for and lead to the following comparisons of the variables :  [ compx]*comparisons of * ) let be as in ( [ garch1 ] ) and ; it follows that .b ) let be as in ( [ garch1 ] ) and ; it follows that .c ) let be as in ( [ garch2 ] ) and ; it follows that .d ) let be as in ( [ garch2 ] ) and ; it follows that .  since and , with independent from , items a ) , and c ) follow from theorem 1.a.3 ( b ) in .similarly item b ) and d ) follow from theorem 4.a.15 in .a natural question that arises at this point is if also the convex order is propagated , that is if .we prove that this is indeed the case for model m1 .we start with a simple lemma :  [ stconv]let and be nonnegative , with .let be independent from and , with = 0 ] , that is . from proposition [ compsigma ] itemb ) , it then follows that . from theorem 4.a.6 in ssthere exists a random variable with . by lemma [ stconv ], we have that implies that ; on the other hand implies that . by transitivitywe get .in the preceding section the orderings defined by , , , have arisen naturally.in order to better understand their meaning , in the following lemmas we identify some necessary and sufficient conditions * * * * in the continuous and symmetric case .we have the following :  let and be symmetric with continuous distributions and .the following conditions are equivalent : a ) ;b ) ;c ) , where is the peakedness ordering introduced in ;d ) with sign sequence , where is the number of intersections between and as defined in ( 1.a.18 ) of .the equivalence of a ) and b ) is an immediate consequence of theorem 1.a.3 in .the equivalence of b ) and c ) is the definition of the peakedness ordering , while the equivalence between c ) and d ) follows from theorem 3.d.1 in .let and be symmetric with continuous distributions and .the following conditions are equivalent : a ) b ) for each , where and ) \leq e[(y^{2}-k)^{+}] ] , and hence \preceq _ { st}e[\widetilde{\varepsilon } _ { k}^{2}] ] .a ) if the densities of and cross times , with the density of being lower in the tails and in the center , and higher in the intermediate region , then .b ) if and and have finite fourth moments , then , where is pearson s kurtosis coefficient .\a ) under our hypothesis for .since and are symmetrical , we have that the four intersection points between the densities and are symmetrical with respect to the origin .hence the densities of and cross in two points and since =e[y^{2}] ] and hence the thesis follows from the definition of the convex order .this lemma shows that the comparison can be interpreted as a classical kurtosis ordering ; the cut condition is usually referred in the kurtosis ordering literature as a dyson - finucan condition ( see for example for a review ) .in financial applications the variables typically represent logreturns , that are additive quantities .the over - the - period total return is given by .it is therefore natural to ask if some of the comparison results of section 2 do extend to the variables . in this sectionwe consider the case of the convex order , that is , we wonder if .the problem is not trivial since can not be expressed as a sum of independent variables , so the standard results about convex ordering of sums can not be applied ; we are able to prove a positive result in the case of model m1 and for symmetric innovations .we start with a basic lemma :  [ bl]let be convex and be convex and nonnegative .let and .it follows that is convex .we can compute first term is positive ; the second is given by us denote with a random vector with a discrete uniform distribution on ; clearly =\underline{0} ] is nondecreasing in the parameters . from the symmetry of the innovations we can write: = e_{\varepsilon _ { 0}, ... ,\varepsilon _ { n}}[\phi ( \sigma _ { 0}\varepsilon _ { 0}+ ... +g_{n}\varepsilon _ { n})]\]]% \text{.}\]]denoting by see that is nondecreasing in ; indeed we can compute : the multivariate covariance inequality , as in the proof of lemma bl .the same reasoning shows that and .it follows that $ ] is is nondecreasing in ; but then if , and , \leq e[\phi ( s_{n}(\tilde{\alpha}_{0},\tilde{\alpha}_{1},\tilde{\beta}_{1})]\]]that is .

Abstract: we consider the problem of stochastic comparison of general garch - like processes , for different parameters and different distributions of the innovations . we identify several stochastic orders that are propagated from the innovations to the garch process itself , and discuss their interpretations . we focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the garch process . more generally , we discuss multivariate comparison results related to the multivariate convex and supermodular order . finally we discuss ordering with respect to the parameters in the garch ( 1,1 ) case . keywords : garch , convex order , peakedness , kurtosis , supermodularity .