Christian H Weiß

Jumps in Binomial AR(1) Processes

Keywords: Binomial AR(1) model, jumps, control chart, average run length

We consider the binomial AR(1) model for serially dependent processes of binomial counts. After a review of its definition and known properties, we investigate marginal and serial properties of jumps in such processes. Based on these results, we propose the jumps control chart for monitoring a binomial AR(1) process. We show how to evaluate the performance of this control chart and give design recommendations.

Introduction

The field of count data time series analysis has become an area of emerging interest both in research and professional practice. Because of the broad field of potential applications, see [START_REF] Weiß | Thinning operations for modelling time series of counts -a survey[END_REF], a number of time series models for counts have been proposed in literature, which are able to describe different types of marginal distribution and autocorrelation structure. In particular, ARMA-type models based on an appropriate type of thinning operation have become quite popular in recent years, see [START_REF] Weiß | Thinning operations for modelling time series of counts -a survey[END_REF]. In this article, we consider the binomial AR(1) model, which was first proposed by [START_REF] Mckenzie | Some simple models for discrete variate time series[END_REF] and further investigated by Weiß (2009a). It is able to model processes of binomial counts with a serial dependence structure being similar to that of a usual AR(1) model. Furthermore, the binomial AR(1) model has an intuitive structure and is easy to interpret.

Definition 1 (Binomial AR(1) Model) Let p ∈ (0; 1) and ρ ∈ max {-p 1-p , -1-p p } ; 1 . Define β := p • (1ρ) and α := β + ρ. Fix n ∈ N.

Email address: christian.weiss@mathematik.uni-wuerzburg.de (Christian H. Weiß).

Preprint submitted to Elsevier

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The process (X t ) N 0 , defined by the recursion

X t = α • X t-1 + β • (n -X t-1 ) for t ≥ 1,
where X 0 ∼ B(n, p), where all thinnings are performed independently of each other, and where the thinnings at time t are independent of (X s ) s<t , is referred to as the binomial AR(1) process.

The condition on ρ guarantees that α, β ∈ (0; 1). The following Summary 1 provides further important properties, which have been shown by [START_REF] Mckenzie | Some simple models for discrete variate time series[END_REF]; Weiß (2009a).

Summary 1 (Properties of Binomial AR(1) Model) Let (X t ) N 0 be a process following the binomial AR(1) model of Definition 1. Then (X t ) N 0 is a stationary Markov chain with marginal distribution B(n, p), i. e.,

p k := P (X t = k) = n k p k (1 -p) n-k ,
and transition probabilities

p k|l := P (X t = k | X t-1 = l) = min {k,l} m=max {0,k+l-n} l m n-l k-m α m (1 -α) l-m β k-m (1 -β) n-l+m-k .
The autocorrelation function is given by ρ(k) = ρ k for k ≥ 0. The most important regression properties are

E[X t | X t-1 ] = ρ • X t-1 + nβ, V [X t | X t-1 ] = ρ(1 -ρ)(1 -2p) • X t-1 + nβ(1 -β).
If X 1 , . . . , X T with T ∈ N is a segment from a stationary binomial AR(1) process, and if N i is the number of X t equal to i and

p i := P (X t = i), then Pearson's statistic X 2 g := n i=0 (N i -T p i ) 2 T p i D → n j=1 1 + ρ j 1 -ρ j • Z 2 j for T → ∞,
where Z 1 , . . . , Z n are independent standard normal random variables.

The binomial AR(1) model of Definition 1 is easy to interpret, see Weiß (2009a). Suppose we have a system of n units, which are, independently of each other, either in state 1 or state 0. Let X t-1 be the number of units being in state 1 at time t -1. Then α Summary 2 (Model Estimation) Let X 0 , . . . , X T be a segment from a binomial AR(1) process with unknown parameters p and ρ, see Definition 1. Weiß (2009a) proposes the following approaches for parameter estimation:

• Yule-Walker approach: Estimate p by the arithmetic mean 1 n(T +1) • T t=0 X t , and ρ by the first order empirical autocorrelation.

• Conditional least squares approach: Estimate p and ρ by minimizing the conditional sum of squares

CSS(p, ρ) := T t=1 X t -ρ • X t-1 -np(1 -ρ) 2 .
• Maximum likelihood approach: Estimate p and ρ by numerically maximizing the logarithmic likelihood function ℓ(p, ρ) := ln p X 0 + T t=1 ln p Xt|X t-1 , see Summary 1.

2 Jumps in Binomial AR(1) Processes

An interesting feature of a binomial AR(1) process are jumps J t := X t -X t-1 . The extent of these jumps reflects the extent of serial dependence, defined through the parameter ρ: If ρ is small, then quite large jumps can be observed.

Theorem 2 (Jumps in Binomial AR(1) Processes: Marginal Properties) Let (X t ) N 0 be a process following the binomial AR(1) model of Definition 1.

Then the moment generating function (mgf ) of (J t ) N is given by

µ J (z) = 1 -2p(1 -p)(1 -ρ) • 1 -cosh (z) n .
All odd order moments of J t are equal to 0. Variance and excess are given by

2σ 2 X • (1 -ρ) and 1 2n • p(1 -p)(1 -ρ) -1 -6 , respectively.
The proof of Theorem 2 is provided by Appendix A.1. Obviously, the variance of the jumps J t increases for decreasing ρ. Hence, see Section 3, we expect a control chart monitoring the jumps J t to be sensitive to decreases in ρ (which would imply that it becomes more likely that a unit leaves the state 1, see Section 1). Furthermore, the distribution of the jumps J t is symmetric around 0 and the excess of J t decreases for increasing n. So the shape of this distribution is similar to that of a normal distribution for large n, a feature that shall prove to be useful for designing a control chart for jumps J t , see Section 3.

Theorem 3 (Jumps in Binomial AR(1) Processes: Serial Properties) Let (X t ) N 0 be a process following the binomial AR(1) model of Definition 1.

Then the autocorrelation and the partial autocorrelation function of (J t ) N are given by

ρ J (k) = - 1 -ρ 2 •ρ k-1 and ρ part,J (k) = - 1 -ρ 2 + (k -1) • (1 -ρ) for k ≥ 1.
The proof of Theorem 3 is provided by Appendix A.2.

An Application: Jumps Control Chart

The discipline of statistical process control (SPC) provides methods for monitoring processes along time. The monitored process (X t ) N 0 is said to be in (statistical) control if it is stationary, following a particular process model. Otherwise, the process is said to be out of control. Assuming that (X t ) N 0 is usually in control, SPC aims at detecting changes in the process that may result from assignable causes at unpredictable times. The most popular tool of SPC is the control chart, which is applied to certain control statistics (Z t ) N computed from the monitored process (X t ) N 0 . A control chart is attached with appropriately chosen upper and lower control limits UCL and LCL. The observations (Z t ) N are plotted on the control chart. If a point Z t exceeds a control limit, i. e., if Z t > UCL or Z t < LCL, then an alarm signals that the process may be out of control. A widely accepted approach to evaluate the performance of a control chart is to compute characteristics of its run length (RL) distribution: The RL of a control chart is the number of plotted points until the first alarm is triggered. In particular, the mean of the RL distribution, the average run length (ARL), is commonly used as a quality factor for the control chart. The chart design is chosen such that the ARL is sufficiently large if the process is in control (the in-control ARL is commonly abbreviated as ARL 0 ), while the ARL should be small if the process is out-of-control. For more details on SPC and control charts, see [START_REF] Montgomery | Introduction to statistical quality control[END_REF].

As outlined in Section 1, the binomial AR( 1) model has a great potential for applications in practice, also see the case study reported by Weiß (2009a). Depending on the concrete situation, the practitioner is interested in monitoring procedures, which allow to detect changes in the model parameters p and/or ρ. Concerning changes in p, especially the moving average charts investigated by Weiß (2009a) turned out to be effective. The jumps computed from a binomial AR(1) process offer the potential for creating a control chart being sensitive to a decrease in ρ, since this would increase the variance of the jumps, see Section 2. Concerning the general interpretation and the examples given in Section 1, such a decrease in ρ would imply that it becomes more likely that a unit leaves the state 1, e. g., that the users of a computer pool more often leave their machines (perhaps indicating technical problems).

Definition and Properties

According to Theorem 2, the stationary distribution of jumps J t in a binomial AR(1) process is symmetric around 0. Hence, it is reasonable to choose symmetric control limits for a control chart monitoring the jumps, i. e., the jumps control chart requires only one design parameter.

Definition 4 (Jumps Control Chart) The observed jumps J t := X t -X t-1 are plotted on a chart with control limits ∓k for a k ∈ N, i. e., the process is considered as being in control unless |J t | > k.

The design parameter k of the jumps control chart 4 should be chosen such that a certain average run length (ARL) performance is reached. The following result is essential for this purpose.

Proposition 5 (Combined Jumps Process) Let (X t ) N 0 be a binomial AR(1) process according to Definition 1. Define the process (J t ) N of jumps by J t := X t -X t-1 . Since (X t ) N 0 is Markovian of order 1, also (X t , J t ) N is a Markov chain, with range {0, . . . , n} × {-n, . . . , n}. The transition probabilities are given by

p(n, j | m, i) := P (X t = n, J t = j | X t-1 = m, J t-1 = i) = P (J t = j | X t = n, X t-1 = m, J t-1 = i) • P (X t = n | X t-1 = m, J t-1 = i) = δ j,n-m • p n|m ,
where δ a,b denotes the Kronecker delta. The marginal probabilities are p(n, j)

:= P (X t = n, J t = j) = P (X t = n, X t-1 = n -j) = p n,n-j ,
where the p k,l := P (X t = k, X t-1 = l) equal 0 for l < 0.

Since at least the bivariate process (X t , J t ) N is a Markov chain, it is possible to compute the ARLs of the jumps chart 4 by adapting the Markov chain approach of [START_REF] Brook | An approach to the probability distribution of cusum run length[END_REF]. For this purpose, it is necessary to identify the reachable in-control states of (X t , J t ) N , i. e., those pairs (x, j) ∈ {0, . . . , n}×{-k, . . . , k}, which can be reached by (X t , J t ) N such that no alarm is triggered by chart 4. It is shown in Appendix A.3 that the set C(k) of such reachable in-control states for given design parameter 0 < k < n is given by

C(k) = (x, j) ∈ {0, . . . , n} × {-k, . . . , k} j ≤ x ≤ n + j , which is of size |C(k)| = (2k + 1)(n + 1) -k(k + 1).
(1)

Using this result and the fact that (X t , J t ) N is a discrete Markov chain according to Proposition 5, the ARLs of the combined jumps chart can be computed exactly by adapting the Markov chain approach of [START_REF] Brook | An approach to the probability distribution of cusum run length[END_REF].

Proposition 6 (ARL Computation of Combined Jumps Chart) Let (X t , J t ) N be the combined jumps process of Proposition 5. Let C := C(k) be the control region of Formula (1). Define p m,i (r) for r ≥ 1 as the probability that the (r+1) st pair leads to the first out-of-control signal, given that (X 1 , J 1 ) = (m, i) ∈ C, i. e., p m,i (r) := P (X r+1 , J r+1 ) ∈ C, (X r , J r ), . . . , (X 2 , J 2 ) ∈ C | (X 1 , J 1 ) = (m, i) , let µ m,i := ∞ r=1 r • p m,i (r). Assume that C is ordered in a certain manner, define µ := (. . . , µ n,j , . . .) ⊤ , Q ⊤ := p(n, j | m, i)

(n,j),(m,i)∈C

.

Then µ is computed as the solution of the linear equation

(I -Q) • µ = 1. The ARL is obtained as ARL = 1 + (m,i)∈C µ m,i • p m,m-i .
The proof of Proposition 6 is done in complete analogy to Weiß (2009b).

Design and Performance

Assume that the parameter values p 0 and ρ 0 of the in-control model of the monitored binomial AR(1) process are available; in practice, these parameters are usually estimated from given in-control data using, e. g., one of the approaches of Summary 2. To design an appropriate jumps chart for the given in-control situation, we have to find a value of k such that a satisfactory ARL 0 is reached. In practice, the desired ARL 0 level is defined by considering aspects like the process speed or the costs related to a false alarm, but in literature, it is common to choose ARL 0 ≈ 370 to make results comparable (The value 370 is motivated by the in-control performance of a standard Shewhart chart with 3-σ limits applied to i.i.d. normal observations.).

To find an appropriate value of the design parameter k, we apply Proposition 6 for the in-control model (p, ρ) = (p 0 , ρ 0 ) and for varying values of k. Based on the computed values ARL 0 (k), we choose k such that ARL 0 (k) is close to the desired in-control performance. The search for an appropriate parameter k can be accelerated by considering that the distribution of jumps has a shape similar to that of a normal distribution, see the discussion after Theorem 2. As an example, if one is interested in reaching an in-control ARL of about 370, then it is known from normal theory (3-σ limits) that the design parameter k should be close to

3 • 2 • np 0 (1 -p 0 ) • (1 -ρ 0 ).
So compared to the conditional and residuals control chart for a binomial AR(1) process proposed by Weiß (2009a), which can also be used for detecting a decrease in ρ, the new jumps control chart is more easy to design and allows to compute ARLs exactly, without the need for simulations.

Next, we study the ARL performance of the jumps chart. Using Proposition 6 for the ARL computations, the performance of the jumps control chart 4 concerning a decrease in ρ compared to the in-control value ρ 0 is investigated. Some of the results are summarized in Figure 1, where the respective incontrol situations (n, p 0 , ρ 0 ) and chart designs k can be read from the legend. The charts considered there are designed in such a way that their in-control performance is similar, reaching an ARL 0 between 300 and 400. It becomes clear that the jumps control chart is sensitive to such a decrease in ρ, with the best sensitivity shown for large values of ρ 0 . Given a particular value of ρ 0 , the sensitivity is better in situations, where n is large and where p 0 has a moderate value (around 0.5).

An Illustrative Example

To illustrate the application of the jumps chart in practice, let us assume that the monitored binomial AR(1) process (X t ) N 0 has the model parameters (n, p, ρ) = (n, p 0 , ρ 0 ) := (25, 0.2, 0.75) in its in-control state. Furthermore, assume that it is desirable to reach an in-control ARL of about 370, which would correspond to 3-σ limits of a standard Shewhart chart applied to i.i.d. normal observations. Adapted to the standard deviation of the jumps, we obtain So we expect that an appropriate value of the design parameter k of the jumps chart will be close to 4. And in fact, we obtain by applying Proposition 6 that . . . , ARL 0 (3) ≈ 64.9961, ARL 0 (4) ≈ 378.012, ARL 0 (5) ≈ 2731.2, . . . so we choose the design parameter k := 4. The theoretical performance of this particular jumps chart concerning a decrease in ρ can be read from Figure 1.

3 • 2 • np 0 (1 -p 0 ) • (1 -ρ 0 ) ≈ 4.24.

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To get an impression of the performance in practice, a segment X 0 , . . . , X 100 from a binomial AR(1) process was simulated in the following way: The observations X 0 , . . . , X 50 stem from the in-control model, but then the parameter ρ is shifted to a value of 0.5. The simulated data is shown in the upper part of Figure 2. The lower part shows the computed jumps J t on a jumps chart with design parameter k = 4 (having the out-of-control ARL ≈ 42.4975 ≪ 378.012, also see Figure 1). While the jumps chart does not signal an alarm during the in-control phase (before the vertical line), the out-of-control situation is detected quickly, with the first alarm being triggered at time t = 55.

Conclusions

We discussed the binomial AR(1) model for serially dependent processes of binomial counts. After having reviewed its definition and known properties, we investigated marginal and serial properties of jumps in such processes. It turned out that the shape of the marginal distribution is similar to that of a normal distribution. This fact is useful for designing the newly proposed jumps control chart for monitoring a binomial AR(1) process. We showed how to compute the ARLs of this control chart and investigated its sensitivity concerning a decrease in ρ compared to the in-control value ρ 0 . Especially for large values of ρ 0 , a decrease in ρ is detected effectively.

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According to [START_REF] Box | Time series analysis -Forecasting and control[END_REF], the a kj of Definition 7 follow the recursive scheme

a k+1,k+1 = ρ(k + 1) - k i=1 a ki • ρ(k + 1 -i) 1 - k i=1 a ki • ρ(i) , a k+1,j = a kj -a k+1,k+1 • a k,k-j+1
for j = 1, . . . , k.

(A.2)

We shall use the recursive scheme (A.2) and prove the following result (by induction):

a kj = - (1 -ρ) • 1 + (k -j)(1 -ρ) 2 + (k -1)(1 -ρ) for 1 ≤ j ≤ k and k ∈ N, (A.3)
with ρ part,J (k) = a kk . For k = 1, we simply obtain ρ part,J (1) = ρ J (1) = -1-ρ 2 , which coincides with (A.3) in this case. So assume that (A.3) has been shown for all k ≤ n with some n ∈ N, consider k = n + 1. We shall use the following recursive scheme, see formula (A.2):

ρ part,J (n + 1) = a n+1,n+1 = ρ J (n+1)- n i=1 a ni •ρ J (n+1-i) 1- n i=1 a ni •ρ J (i) , a n+1,j = a nj -a n+1,n+1 • a n,n+1-j for j = 1, . . . , n.
Applying the geometric sum formulae, one obtains the following results:

n i=1 a ni • ρ J (n + 1 -i) = (1-ρ) 2
2 2+(n-1)(1-ρ)

• n i=1 1 + (n -i)(1 -ρ) • ρ n-i = 1-ρ 2 2+(n-1)(1-ρ) • 1 + ρ -(n + 1) • ρ n + (n -1) • ρ n+1 , n i=1 a ni • ρ J (i) = (1-ρ) 2
2 2+(n-1)(1-ρ)

• n i=1 1 + (n -i)(1 -ρ) • ρ i-1 = n(1-ρ) 2 2 2+(n-1)(1-ρ)
.

Therefore, it results that

ρ J (n + 1) -n i=1 a ni • ρ J (n + 1 -i) = -(1-ρ 2 ) 2 2+(n-1)(1-ρ)
, and 1 -n i=1 a ni • ρ J (i) = • 1 + (nj)(1ρ)

• 2 + n(1 -ρ) + (1 -ρ) • 1 + (j -1)(1 -ρ) = -(1-ρ) 2+(n-1)(1-ρ) • 2+n(1-ρ) • 2 + (1 -ρ) • 2(n -j) + n + 1 + (1 -ρ) 2 • n(n -j) + j -1 identity = -(1-ρ) 2+(n-1)(1-ρ) • 2+n(1-ρ) • 2 + (n -1)(1 -ρ) • 1 + (n + 1 -j)(1 -ρ) ,
which completes the proof.

A.3 Proof of Formula (1) Since J t = X t -X t-1 , we have J t = j iff X t = j + X t-1 . Since 0 ≤ X t-1 ≤ n, the restriction j ≤ X t ≤ n + j follows

Fig. 1 .Fig. 2 .

 12 Fig. 1. ARL performance of jumps charts for different in-control models.

(

  expression for a n+1,n+1 = -(1ρ) 2 + n(1ρ) follows. Next, one shows the identity 2 + (1ρ) • 2(nj) + n + 1 + (1ρ) 2 • n(nj) + j -1 = 2 + (n -1) • (1ρ) • 1 + (n + 1j) • (1ρ)by simply multiplying out the product on the right-hand side and collecting the resulting terms. Using this identity, one obtains a n+1,j = a nja n+1,n+1 • a n,n+1-j

  . The size |C(k)| follows from evaluating the sum|C(k)| = k j=-k min {n, n + j} -max {0, j} + n -1 2 k(k + 1) + (k + 1) • n -0 + 1 2 k(k + 1) + (2k + 1) = (2k + 1) • nk(k + 1) + (2k + 1) = (2k + 1)(n + 1)k(k + 1).

Acknowledgements. The author thanks the referee for useful comments on an earlier draft of this article.

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 Weiß, C.H., 2009b. Controlling Jumps in Correlated Processes of Poisson Counts.Appears in Applied Stochastic Models in Business and Industry 25.

A Proofs

A.1 Proof of Theorem 2

Let (X t ) N 0 be a process following the binomial AR(1) model of Definition 1 with parameters n, ρ, p. Then the bivariate probability generating function (pgf) p Xt,X t-1 (z 0 , z 1 ) of X t and X t-1 is obtained as

Using the expression (A.1) for the bivariate pgf, we obtain the mgf of (J t ) N . Here, we use the relations

Since the series expansion of cosh (z) contains only even terms, the same is true for µ J (z), i. e., all odd order moments of J t are equal to 0. Even order moments are obtained from the relation µ J

Since cosh ′ (z) = sinh (z) and sinh ′ (z) = cosh (z), we have

Since cosh (0) = 1 and sinh (0) = 0, it follows that

A.2 Proof of Theorem 3

Since

), which proves the expression for ρ J (k). So it remains to prove the expression for the partial autocorrelation function. For this purpose, we adopt the notations of the following Definition 7: Definition 7 (Partial Autocorrelation) Let (X t ) Z be a weakly stationary process with 0 < γ(0) < ∞, denote ρ(k) := Corr[X t , X t-k ] for k ∈ N 0 , with ρ(0) = 1. For k ∈ N, define the matrix R k := ρ(|i -j|) 1≤i,j≤k .

For each k ∈ N, we define (a k1 , . . . , a kk ) ⊤ as the solution of the equation R k (a k1 , . . . , a kk ) ⊤ = ρ(1), . . . , ρ(k)

⊤ .

Then ρ part (k) := a kk is referred to as the k th order partial autocorrelation.


Tasos C Christofides

Milto Hadjikyriakou
email: miltwh@gmail.com

Exponential inequalities for n-demimartingales and negatively associated random variables

Keywords: Negatively associated random variables, Demimartingales, N-demimartingales, Azuma's inequality, Exponential inequalities, Complete convergence AMS 2000 subject classification: 60E15, 60G48

. In this paper we prove Azuma's inequality for N-demimartingales and as a corollary we obtain an exponential inequality for negatively associated random variables.

Introduction

In recent years, dependence concepts including positive and negative association introduced by [START_REF] Esary | Association of random variables with applications[END_REF] and Joag-Dev and Proschan (1983) respectively have been the focus of substantial research activity. Among the various results presented are extensions and generalizations. In particular, [START_REF] Newman | Associated random variables and martingale inequalities[END_REF] introduced the concept of a demimartingale and a demisubmartingale as a generalization of the notion of martingales and submartingales. The definition is a rather technical one and serves, among other things, the purpose of studying in a more general way the behavior of the partial sum of mean zero associated random variables.

Definition 1 A sequence of L 1 random variables {S n , n ∈ N} is called a demimartingale if for all j = 1, 2, . . . E [(S j+1 -S j )f (S 1 , . . . , S j )] ≥ 0

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for all componentwise nondecreasing functions f whenever the expectation is defined. Moreover, if f is assumed to be nonnegative, the sequence {S n , n ∈ N} is called a demisubmartingale.

Definition 2 A finite collection of random variables X 1 , . . . , X n is said to be (positively)associated if

Cov(f (X 1 , . . . , X n ), g(X 1 , . . . , X n )) ≥ 0
for any componentwise nondecreasing functions f, g on R n such that the covariance is defined. An infinite collection is associated if every finite subcollection is associated.

One can easily verify that the partial sum of mean zero positively associated random variables is a demimartingale.

Motivated by the definition of a demimartingale, the idea of a similar generalization for negatively associated random variables leads to the concept of the so called Ndemimartingales and N-demisupermartingales. Definition 3 A sequence of L 1 random variables {S n , n ∈ N} is called an N-demimartingale if for all j = 1, 2, . . . E [(S j+1 -S j )f (S 1 , . . . , S j )] ≤ 0 for all componentwise nondecreasing functions f provided the expectation is defined. Moreover, if f is assumed to be nonnegative, the sequence {S n , n ∈ N} is called an Ndemisupermartingale.

Various results and examples of N-demimartingales and N-demisupermartingales can be found in [START_REF] Christofides | Maximal Inequalities for N-demimartingales[END_REF] and Prakasa [START_REF] Rao | On some inequalities for N-demimartingales[END_REF][START_REF] Rao | On some maximal inequalities for demisubmartingales and N-demisupermartingales[END_REF].

Let us now recall the following definition of negative association. Definition 4 A finite collection of random variables X 1 , . . . , X n is said to be negatively associated if

Cov(f (X i , i ∈ A), g(X j , j ∈ B)) ≤ 0
for any disjoint subsets A and B of {1, 2, . . . , n} and for any two componentwise nondecreasing functions f, g on R A and R B respectively, provided that the covariance is defined. An infinite collection is negatively associated if every finite subcollection is negatively associated.

It is trivial to verify that the partial sum of mean zero negatively associated random variables is an N-demimartingale.

It is worth mentioning that a martingale with the natural choice of σ-algebras is both a demimartingale and an N-demimartingale. Furthermore, it can be verified that a subis a demisubmartingale, and a supermartingale is an N-demisupermartingale.

In this paper, we prove Azuma's inequality for N-demimartingales and as a corollary, we obtain an exponential inequality for the partial sum of mean zero negatively associated random variables. Finally, complete convergence results for N-demimartingales and negatively associated random variables are presented.

Azuma's inequality for N-demimartingales

The following result for martingale differences was given by [START_REF] Azuma | Weighted sums of certain dependent random variables[END_REF].

Theorem 5 Let {X n , n ∈ N} be a sequence of martingale differences such that |X i | < α < ∞ for all i = 1, 2, . . . and let S n = X 1 + • • • + X n . Then for every ε > 0, P (S n ≥ nε) ≤ exp -nε 2 2α 2 .
Given that a martingale with the natural choice of σ-algebras is an N-demimartingale, it is of interest to see whether an analog of the above inequality holds true for Ndemimartingales. The answer is given by the following result.

Theorem 6 Let {S n , n ∈ N} (with S 0 ≡ 0) be an N-demimartingale and assume that

|S i -S i-1 | ≤ c i < ∞ i = 1, 2, . . . ,
where c 1 , c 2 , . . . are positive real numbers. Then for every ε > 0,

P (S n ≥ nε) ≤ exp -n 2 ε 2 2 n i=1 c 2 i ( 1
)
and

P (|S n | ≥ nε) ≤ 2 exp -n 2 ε 2 2 n i=1 c 2 i . (2)
Proof. Let t ∈ R and x ∈ [-c i , c i ]. We can write

tx = 1 2 1 + x c i (c i t) + 1 2 1 - x c i (-c i t).
By the convexity of the exponential function we have:

e tx ≤ cosh(c i t) + x c i sinh(c i t).

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Then

E e tSn = E n i=1 e t(S i -S i-1 ) ≤ E n i=1 cosh(c i t) + (S i -S i-1 ) sinh(c i t) c i .
By induction we will prove that

E e tSn ≤ n i=1 cosh(c i t) ∀ t > 0. ( 3
)
First observe that,

E e tS2 ≤ E cosh(c 1 t) + S 1 sinh(c 1 t) c 1 cosh(c 2 t) + (S 2 -S 1 ) sinh(c 2 t) c 2 = cosh(c 1 t) cosh(c 2 t) + sinh(c 1 t) sinh(c 2 t) c 1 c 2 E[S 1 (S 2 -S 1 )] ≤ cosh(c 1 t) cosh(c 2 t)
where the last inequality follows from the N-demimartingale property. Thus (3) is true for n = 2. Assume now that the statement is true for n = k. We will show that it is true for n = k + 1.

E e tS k+1 = E e t(S k+1 -S k ) .e tS k ≤ E cosh(c k+1 t)

+ (S k+1 -S k ) sinh(c k+1 t) c k+1 e tS k = cosh(c k+1 t)E e tS k + sinh(c k+1 t) c k+1 E (S k+1 -S k )e tS k ≤ k+1 i=1 cosh(c i t)
where the last inequality follows from the N-demimartingale property and the induction hypothesis. Thus (3) is established.

Since cosh(c i t) ≤ e c 2 i t 2 2 , by inequality (3) we have:

E e tSn ≤ exp t 2 n i=1 c 2 i 2 .
For ε, t > 0

P (S n ≥ nε) = P (tS n ≥ tnε) = P (e tSn ≥ e tnε ) ≤ e -tnε E e tSn ≤ exp -tnε + t 2 n i=1 c 2 i 2 .
The above upper bound is minimized by choosing t = nε/ n i=1 c 2 i and ( 1) is established. To prove inequality (2) we write:

P (|S n | ≥ nε) = P (S n ≥ nε) + P (-S n ≥ nε).
Since the collection {-S n , n ∈ N} is also an N-demimartingale (see Remark 1.4 in Christofides ( 2003)), inequality (2) follows by applying inequality (1) twice.

Remark 7

In case the N-demimartingale differences are uniformly bounded, i.e., when 1) and ( 2) have the simple form

|S i -S i-1 | ≤ c < ∞ for i = 1, 2, . . ., then (
P (S n ≥ nε) ≤ exp -nε 2 2c 2 ,
and

P (|S n | ≥ nε) ≤ 2 exp -nε 2 2c 2
respectively.

Applications

Given that the partial sum of mean zero negatively associated random variables is an N-demimartingale, we immediately have the following result.

Corollary 8 Let {X n , n ∈ N} be mean zero negatively associated random variables such that

|X k | ≤ c k k = 1, 2, . . .. Let S n = X 1 + • • • + X n .
Then for every ε > 0,

P (S n ≥ nε) ≤ exp -n 2 ε 2 2 n i=1 c 2 i , and
P (|S n | ≥ nε) ≤ 2 exp -n 2 ε 2 2 n i=1 c 2 i .
Remark 9 Exponential inequalities for mean zero negatively associated random variables {X i , i ≥ 1} have been obtained by [START_REF] Han | Exponential inequality and almost sure convergence for the negatively associated sequence[END_REF] following a different approach, and in particular a Hoeffding inequality under the assumption that a i ≤ X i ≤ b i for all i. It is worth mentioning that for b i = c i and a i = -c i for i = 1, 2 . . ., Corollary 8 provides a much sharper bound.

As an application of Corollary 8 we give the following result which was first proven by Matula (1996).

Corollary 10 Let {X n , n ∈ N} be a sequence of negatively associated random variables with common distribution function F . Then for every ε > 0 and ∀ x ∈ R,

P (|F n (x) -F (x)| > ε) ≤ 2 exp -nε 2 2
where F n denotes the empirical distribution function.

The proof of Corollary 10, the details of which are omitted for brevity, is based on the observation that for fixed x, {I {Xn≤x} , n ∈ N} is a sequence of negatively associated random variables satisfying

|I {Xn≤x} -F (x)| ≤ 1, n = 1, 2, . . . ,
where I {Xn≤x} is the indicator function of the set {X n ≤ x}.

The following exponential inequality for mean zero negatively associated random variables is due to Matula (1996).

Theorem 11 Let {X n , n ∈ N} be mean zero negatively associated random variables such that

|X k | ≤ α ∀ k. Let S n = X 1 + • • • + X n .
Then for every ε > 0,

P (S n ≥ nε) ≤ exp -nε 2α sinh -1 nεα 2b n .
where b n = n i=1 EX 2 i .

The bounds of Corollary 8 (for the special case where c k = α for k = 1, 2, . . .) and Theorem 11 are not directly comparable given that the bound in Theorem 11 is expressed in terms of the variances of the random variables. Let y = ε/α. To compare the two bounds it is sufficient to check under what conditions the function The next two asymptotic results concern the complete convergence for bounded Ndemimartingales.

f (y) = y -sinh -1 nyα 2 2b n ,
Theorem 12 Let {S n , n ∈ N} be an N-demimartingale such that | S i -S i-1 |≤ α for all i = 1, 2, . . .. Then for r > 1 2 , n -r S n -→ 0 completely.

Proof.

∞ n=1 P (| S n |≥ n r ε) ≤ 2 ∞ n=1 exp -n 2r-1 ε 2 2α 2 = 2 ∞ n=1 exp -n 2r-1 d = 2 ∞ n=1 (exp(-d)) n 2r-1 < ∞
where d = ε 2 /2α 2 and the first inequality follows from Azuma's inequality for Ndemimartingales given in Remark 7.

Theorem 13 Let {S n , n ∈ N} be an N-demimartingale such that | S i -S i-1 |≤ c i , for all i = 1, 2, . . .. Assume that ∞ i=1 c 2 i < ∞. Then for r > 0, n -r S n -→ 0 completely.

Proof.

∞ n=1 P (| S n |≥ n r ε) ≤ 2 ∞ n=1 exp -n 2 n 2r-2 ε 2 2 n i=1 c 2 i = 2 ∞ n=1 exp -n 2r ε 2 2 n i=1 c 2 i ≤ 2 ∞ n=1 exp -n 2r ε 2 2 ∞ i=1 c 2 i = 2 ∞ n=1 (exp(-d)) n 2r < ∞
where d = ε 2 /2 ∞ i=1 c 2 i and the first inequality follows from inequality [START_REF] Christofides | Maximal Inequalities for N-demimartingales[END_REF].

As an application of Theorem 12 we immediately have the following asymptotic result for mean zero negatively associated random variables.

Corollary 14 Let {X n , n ∈ N} be mean zero negatively associated random variables such that |X k | ≤ α for k = 1, 2, . . .. Let S n = X 1 + • • • + X n . Then for r > 1 2 , n -r S n -→ 0 completely.

  y > 0 is nonnegative. Simple calculations show that if nα 2 ≤ 2b n or y ≥ 1 then f is nondecreasing and since f (0) = 0, the function is nonnegative. This means that if nα 2 ≤ 2b n or ε ≥ α the bound of Corollary 8 is sharper than the bound of Theorem 11. However, if nα 2 > 2b n and y < α 4 -4b 2 n /n 2 , i.e., if nα 2 > 2b n and ε < α -1 α 4 -4b 2 n /n 2 , the function is nonincreasing implying that Matula's bound is sharper than the bound of Corollary 8.


Marco Di Marzio
email: mdimarzio@dmqte.unich.it

Agnese Panzera
email: agnesepanzera@yahoo.it

Charles C Taylor
email: charles@maths.leeds.ac.uk

Local polynomial regression for circular predictors

Keywords: Circular data, circular kernels, von Mises weight function, weighted least squares. 2000 MSC: 62G07 -62G08 -62G20

come

Introduction

A circular observation can be regarded as a point on the unit circle, or a direction in the plane. Once an initial direction and an orientation of the unit circle have been chosen, any circular observation may be represented by an angle θ ∈ [0, 2π). Typical examples include flight direction of birds from a point of release, wind and ocean current direction, energy demand over a period of 24 hours when the measurements are taken over a time interval much longer than the day and when the times of the day are recorded. A circular observation is periodic, i.e., θ = θ + 2mπ for m ∈ . This periodicity sets apart circular statistical analysis from standard real-line methods. Recent accounts are given by [START_REF] Jammalamadaka | Topics in Circular Statistics[END_REF] and [START_REF] Mardia | Directional Statistics[END_REF].

A much less studied subject is local regression in the case of circular predictors and real-valued responses. Its practical relevance is easily seen when considering the analysis of meteorological data, or more generally in earth and environmental sciences. Silverman (1986, sec. 2.10) suggests fitting data replicated along the interval [-2π, 4π), with a smoothing degree depending on the original sample size. The only alternative approach appears to be periodic smoothing splines, introduced by [START_REF] Cogburn | Periodic splines and spectral estimation[END_REF]. Nothing specific and reasonably simple appears to exist for the high-dimensional case, although this seems needed in many applications. For example, it could be of interest to predict ozone concentration given the wind directions at 6am and at noon. In this example, the number of angles is d = 2, but this could easily be extended by considering more locations or time points for the explanatory wind directions; see Mardia & Jupp (1999, pp. 1-12) for further examples.

In this paper we extend least squares local polynomial fitting [START_REF] Ruppert | Multivariate locally weighted least squares regression[END_REF], for example) to the case when a design point θ is a vector of angles (θ 1 ,

• • • , θ d ) T ∈ [0, 2π) d ,
and the response is real-valued. Geometrically, θ identifies a point of a d-dimensional torus made of the cartesian product of d unit circles. Our strategy is twofold. We i) introduce a class of circular weight functions (or kernels), and ii) locally approximate the design density and the regression function by the pth degree polynomial

β 0 + d j=1 p t=1 β jt sin t (• -θ j ).
(1)

Point ii) is motivated by the fact that the difference between two angular observations needs to be minimal at 2mπ, m ∈ . Moreover, because sin(θ) ⋍ θ as θ tends to 0, the polynomial (1) satisfies a Taylor series interpretation.

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In Section 2 we define the kernels suitable for our polynomial fitting, and explore their efficiency properties. In Section 3 we consider the local linear (p = 1) regression estimator, along with conditional mean squared error and optimal smoothing. We also extend the analysis, for univariate predictors, to general p. Finally, Section 4 contains a small simulation study to illustrate the finite sample behaviour of the results.

Circular kernels

Definitions

We introduce our kernels in the one-dimensional setting. Such an approach seems adequate in that we will use as weight functions products of univariate kernels, as the torus geometry allows for.

Definition 1. (Circular kernels of order r) A circular kernel, of order r and concentration (smoothing) parameter

κ > 0, is a function K κ : [0, 2π) → Ê such that i) it admits, at θ ∈ [0, 2π), a convergent Fourier series representation 1/(2π){1 + 2 ∞ j=1 γ j (κ) cos( jθ)}; ii) denoting η j (K κ ) := 2π 0 sin j (θ)K κ (θ)dθ , then η 0 (K κ ) = 1 , η j (K κ ) = 0 for 0 < j < r , and η r (K κ ) 0 ;
iii) as κ increases ǫ -ǫ K κ (θ)dθ tends to 1 for ǫ ∈ (0, π) . Condition i) specifies that the kernel is symmetric around the null mean direction. The quantity η j (K κ ) in ii) plays a similar rôle as the jth moment of a symmetric kernel in the linear theory, being zero if j is odd.

Remark 1. Most of the usual circular densities, which are symmetric about the null mean direction, are included in Definition 1 as second-order kernels -this includes the kernel uniform on [-π Remark 2. Our order definition is consistent with the techniques used for obtaining higher order kernels starting from second-order ones. As an instance, we apply a technique of [START_REF] Lejeune | Smooth estimators of distribution and density functions[END_REF], to get a result useful in Theorem 4. Given a second-order circular kernel K κ , let E ℓ be a matrix of order ℓ + 1 with (i, j) entry given by η i+ j-2 (K κ ), and U ℓ be the same as E ℓ with the first column replaced by {1, sin(θ), • • • , sin ℓ (θ)} T . Then

K (ℓ) (θ) := |U ℓ | |E ℓ | K κ (θ) ,
is a circular kernel of order ℓ + 1 when ℓ is odd, and of order ℓ + 2 otherwise.

Remark 3. The univariate setting allows for a comparison with previous work. Our kernels include kernels on the sphere which are functions of κ{1cos(θ)} studied by [START_REF] Beran | Exponential models for directional data[END_REF], [START_REF] Hall | Kernel density estimation with spherical data[END_REF], [START_REF] Bai | Kernel estimators of density function of directional data[END_REF] and [START_REF] Klemelä | Estimation of densities and derivatives of densities with directional data[END_REF]. However, the kernels D κ , F κ and the wrapped Cauchy are not of this latter form, yet fulfil the conditions of Definition 1.

Kernel efficiency

We discuss the efficiency of our kernels in the density estimation setting to allow easy comparisons with the standard theory.

Definition 2. (Kernel circular density estimator)

Let Θ 1 , • • • , Θ n be a random sample from a bounded, continuous circular density f . Given a circular kernel K κ , the kernel estimator of f at θ ∈ [0, 2π) is defined as

f (θ; κ) := 1 n n i=1 K κ (θ -Θ i ) . (2)
The efficiency theory of euclidean kernels (p. 42 Silverman 1986, for example) is based on the fact that the bandwidth and the kernel have separable contributions to the mean integrated squared error

MISE[ ĝ] := E[( ĝ-g) 2 ] ≡ (E[ ĝ] -g) 2 + Var[ ĝ]
, where ĝ gives the kernel estimate of the curve g at a point of the domain. Unfortunately, this is not the case for the MISE of (2). In fact, we have

Theorem 1. Given a random sample Θ 1 , • • • , Θ n drawn from a density f , let f (• ; κ) be the kernel circular density estimator equipped with the second-order kernel K κ , if i) lim n→∞ γ j (κ) = 1, for each j ∈ + ; ii) lim n→∞ n -1 ∞ j=1 γ 2 j (κ) = 0; iii) f ′′ is continuous and square-integrable; then MISE f (• ; κ) = 1 16 {1 -γ 2 (κ)} 2 2π 0 f ′′ (θ) 2 dθ + 1 + 2 ∞ i=1 γ 2 j (κ) 2nπ + o(1) , Proof. See Appendix.
Remark 4. The MISE of [START_REF] Hall | Kernel density estimation with spherical data[END_REF] is very similar to that above. For example, consider the von Mises kernel, for which γ j (κ) := I j (κ)/I 0 (κ), I j (•) being the modified Bessel function of the first kind and order j. Using the notation of (3.7) in [START_REF] Hall | Kernel density estimation with spherical data[END_REF], we have:

c 2 0 (κ)c 2 (κ) = I 0 (2κ)/[2π{I 0 (κ)} 2 ] = {1 + 2 ∞ i=1 γ 2 j (κ)}/(2π) and 1 -c 0 (κ)c 1 (κ) = 1 -I 1 (κ)/I 0 (κ) = 1 -γ 1 (κ)
, consequently their asymptotic MISE differs from the leading terms in the above MISE of an order of O(κ -4 ) .

In our efficiency analysis we need

Result 1. Let Θ 1 , • • • , Θ n be a random sample from a circular density f having Fourier series expansion f (θ) = 1/(2π)[1 + 2 ∞ j=1 {α j cos( jθ) + δ j sin( jθ)}] for θ ∈ [0, 2π). Then MISE f (• ; κ) = 1 π ∞ j=1 {γ j (κ) -1} 2 (α 2 j + δ 2 j ) + 1 nπ ∞ j=1 γ 2 j (κ)(1 -α 2 j -δ 2 j ) .
Without loss of generality we can suppose that the mean direction is 0, and we consider only densities and kernels which are fully specified by their concentration parameters, respectively denoted as ρ and κ. For the above decomposition, when considering the (relative) efficiency of two circular kernels, the smoothing parameters do not "cancel" and so their equivalence needs first to be established as follows. For fixed ρ and n, we can obtain κ to minimize MISE for a given kernel function. The efficiency of one kernel relative to another may then be measured by taking the ratio of the minimized MISEs.

As the Dirichlet kernel (γ j (κ) = ½ { j≤κ} ) is of higher order for κ > 1 -and so expected to be asymptotically more efficient -we have measured the efficiency of other kernels relative to this one. In Figure 1 we show the relative efficiency of the von Miseswrapped normal (γ j (κ) = κ j 2 ), and Fejér (γ j (κ) = ½ { j≤κ} (κ + 1 -j)/(κ + 1)) kernels for n = 5, 25, 125, 625 for the von Mises and wrapped Cauchy (α j = ρ j ; δ j = 0) distributions. Not surprisingly, the wrapped Normal and von Mises kernels are very similar, and both are better than the Fejér kernel. For small n, the von Mises kernel is more efficient that the Dirichlet kernel; markedly so for the Cauchy distribution, or for data with low concentration. where σ 2 (•) is the conditional variance of Y and ε i s are real-valued random variables with zero mean and unit variance.

Local polynomial regression

Linear fitting with von Mises based kernels

Consider the dataset {(Θ i , Y i ), i = 1, • • • , n}, where Θ i := (Θ i1 , • • • , Θ id ) T ,
Y i = m(Θ i ) + σ(Θ i )ε i , i = 1, • • • , n
Our objective is to construct an estimator of m(θ) as a function of the dataset when both Θ i s and ε i s are i.i.d.. Let P θ (•; β) := β 0 + d j=1 β j sin(•θ j ), and suppose that m(ψ) ≃ P θ (ψ; β) for ψ in a neighborhood of θ. Here P θ (θ; β) = β 0 , which motivates estimating m(θ) by β0 . Recalling that for very small values of θ we have sin(θ) ≃ θ, then a Taylor series expansion justifies both β0 and the values β j , j = 1, • • • , d, as estimates of the partial derivatives β j = ∂m(θ)/∂θ j . Viewed as local least squares estimators, β0 , • • • , βd minimize n i=1 {Y i -P θ (Θ i ; β)} 2 w(Θ i , θ) where w(Θ i , θ) is the weight function, (a symmetric, continuous function integrating to 1) which, if strictly positive, decreases with some distance between Θ i and θ. Now we provide an explicit expression for β0 together with its L 2 properties.

Let

y := (Y 1 , • • • , Y n ) T be the response vector, Θ :=             1 sin(Θ 11 -θ 1 ) • • • sin(Θ 1d -θ d ) . . . . . . . . . . . . 1 sin(Θ n1 -θ 1 ) • • • sin(Θ nd -θ d )            
the design matrix, and

W := diag {K C (Θ 1 -θ), • • • , K C (Θ n -θ)}
the weight matrix, where C := κI, I denoting the identity matrix of order d, and

K C (Θ i -θ) := d j=1 K κ (Θ i j -θ j ) , i = 1, • • • , n. ( 3
)
The local linear kernel estimator of m(θ) is given by the first entry of the vector β := arg min

β n i=1 (Y i -β T Θ) 2 K C (Θ i -θ) ,
where

β := (β 0 , β 1 , • • • , β d ) T .
Assuming the non-singularity of Θ T WΘ, standard weighted least squares theory yields β = (Θ T WΘ) -1 Θ T Wy, and m(θ;

C) = e T j (Θ T WΘ) -1 Θ T Wy , (4)
where e j is a (d + 1) × 1 vector having 1 as the jth entry and 0 elsewhere. Given its efficiency, as well as its prevalence in kernel smoothing of circular data, we firstly give results when the von Mises kernel V κ (•) := exp{κ cos(•)}/{2πI 0 (κ)} is used to define the d-dimensional weight function.

Theorem 2. Given the dataset {(Θ i , Y i ), i = 1, • • • , n},
(Θ i -θ) := d j=1 V κ (Θ i j -θ j ). Assume that i) lim n→∞ κ -1 = 0; ii) lim n→∞ n -1 κ d/2 = 0;
iii) the conditional variance σ 2 is continuous, and the density f is continuously differentiable; iv) all second-order derivatives of the regression function m are continuous.

Then at θ ∈ [0, 2π) d the conditional mean squared error of m(θ ; C) is given by

E[{ m(• ; C) -m(θ)} 2 | Θ 1 , • • • , Θ n ] = 1 4 I 1 (κ) κI 0 (κ) 2 tr 2 {H m (θ)} + I 0 (2κ) 2π{I 0 (κ)} 2 d σ 2 (θ) n f (θ) + o p κ -2 + n -1 κ d/2 , ( 5
)
where H m (θ) denotes the Hessian matrix of m at θ.

Proof. See Appendix.

Once more, in the proof of the above theorem a major technical issue is that the concentration parameter κ cannot be "separated" from the kernel.

Remark 5. Since κ corresponds to the inverse of the squared bandwidth of the euclidean smoother, the remainder term in ( 5) is consistent with that obtained by [START_REF] Ruppert | Multivariate locally weighted least squares regression[END_REF].

Finally, the optimal smoothing degree is given by Corollary 1. The concentration parameter which minimizes the asymptotic mean squared error, i.e. the first two summands in RHS of formula ( 5), is

tr 4 {H m (θ)}{n f (θ)} 2 2 2d π d d 2 σ 4 (θ) 1/(4+d)
.

Proof. See Appendix.

Generalizations and extensions

The results of Theorem 2 can be generalized to the class of second-order circular kernels K κ . Given the squareintegrable function g, define R(g) := g 2 , then

Theorem 3. Given the dataset {(Θ i , Y i ), i = 1, • • • , n}, where Θ i s are i.i.d.

observations from the circular design density f , and Y i s are i.i.d. real-valued random variables, take the local linear kernel regression estimator m(• ; C) equipped with the weight function in (3) with K κ being a second-order circular kernel. Assume conditions i) of Theorem 1, and iii) of Theorem 2, together with

i) lim n→∞ n -1 R(K C ) = 0. Then, at θ ∈ [0, 2π) d , E[{ m(• ; C) -m(θ)} 2 | Θ 1 , • • • , Θ n ] = 1 16 {1 -γ 2 (κ)} 2 tr 2 {H m (θ)} + R(K C )σ 2 (θ) n f (θ) + o p (1).
Proof. See Appendix.

It would be of interest to determine the optimal smoothing degree in this case, but since the coefficients γ j s depend on κ in a specific way for each kernel, the result in Corollary 1 is hard to generalize. Concerning the extension to higher-degree polynomials and whatever second-order circular kernel, we have

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Now, recalling assumptions i) and ii), we have

Var[ f (θ; κ)] = 1 n 2π 0 {K κ (ψ -θ)} 2 f (ψ)dψ - 1 n E[ f (θ; κ)] 2 = 1 n 2π 0 {K κ (u)} 2 { f (θ) + o(1)}du - 1 n { f (θ) + o(1)} 2 = 1 2nπ          1 + 2 ∞ j=1 γ 2 j (κ)          f (θ) + o(1) . Proof of Theorem 2. Put S Θi-θ := {sin(Θ i1 -θ 1 ), • • • , sin(Θ id -θ d )} T , i = 1, • • • , n
and use D g (θ) to denote the first-order partial derivatives vector of the function g at θ. To derive the conditional bias, we firstly note that ( 4) yields

E[ m(θ; C) | Θ 1 , • • • , Θ n ] = e T 1 (Θ T WΘ) -1 Θ T Wm , (6)
where

m := {m(Θ 1 ), • • • , m(Θ n )} T , and W := diag {V C (Θ 1 -θ), • • • , V C (Θ n -θ)}. Using the expansion m = Θ m(θ) D m (θ) + 1 2              S T Θ1-θ H m (θ)S Θ1-θ . . . S T Θn-θ H m (θ)S Θn-θ              + R m (θ) ,
where R m (θ) denotes the remainder, we have that the first term in the expansion of ( 6) is m(θ). Thus

E[ m(θ; C) -m(θ) | Θ 1 , • • • , Θ n ] = 1 2 e T 1 (Θ T WΘ) -1 Θ T W                           S T Θ1-θ H m (θ)S Θ1-θ . . . S T Θn-θ H m (θ)S Θn-θ              + R m (θ)              .
Observe that

Θ T WΘ = n i=1 V C (Θ i -θ) n i=1 V C (Θ i -θ)S T Θi-θ n i=1 V C (Θ i -θ)S Θi-θ n i=1 V C (Θ i -θ)S Θi-θ S T Θi-θ (7) and Θ T W              S T Θ1-θ H m (θ)S Θ1-θ . . . S T Θn-θ H m (θ)S Θn-θ              =       n i=1 V C (Θ i -θ)S T Θi-θ H m (θ)S Θi-θ n i=1 V C (Θ i -θ) S T Θi-θ H m (θ)S Θi-θ S Θi-θ       , (8)
then, using the expansion

f (u + θ) = f (θ) + S T u D f (θ) + O(S T u S u )
, and recalling assumption i), a change of variables leads to these approximations

1 n n i=1 V C (Θ i -θ) = [0,2π) d V C (α -θ) f (α)dα + o p (1) = f (θ) + o p (1); 1 n n i=1 V C (Θ i -θ)S Θi-θ = [0,2π) d V C (α -θ)S α-θ f (α)dα + o p (1) = I 1 (κ) I 0 (κ) C -1 D f (θ) + o p (C -1 1) ; 1 n n i=1 V C (Θ i -θ)S Θi-θ S T Θi-θ = [0,2π) d V C (α -θ)S α-θ S T α-θ f (α)dα + o p (I) = I 1 (κ) I 0 (κ) C -1 f (θ) + o p (C -1 ); 1 n n i=1 V C (Θ i -θ)S T Θi-θ H m (θ)S Θi-θ = [0,2π) d V C (α -θ)S T α-θ H m (θ)S α-θ f (α)dα + o p (1) = I 1 (κ) κI 0 (κ) tr {H m (θ)} f (θ) + o p κ -1 ; 1 n n i=1 V C (Θ i -θ) S T Θi-θ H m (θ)S Θi-θ S Θi-θ = [0,2π) d V C (α -θ) S T α-θ H m (θ)S α-θ S α-θ f (α)dα + o p (1) = O p (C -2 1) ;
where 1 is the unit vector of length d. Hence, recalling assumption i) we have

e T 1 n -1 Θ T WΘ -1 ⋍ { f (θ)} -1 + o p (1) -D f (θ) T { f (θ)} -2 + o p (1) , (9) thus
E[ m(θ; C) -m(θ) | Θ 1 , • • • , Θ n ] = 1 2 I 1 (κ) κI 0 (κ) tr {H m (θ)} + o p (κ -1 ) .
For the conditional variance, according to multivariate local linear regression theory

Var[ m(θ; C) | Θ 1 , • • • , Θ n ] = e T 1 (Θ T WΘ) -1 Θ T WΣWΘ(Θ T WΘ) -1 e 1 ,
where

Σ := diag {σ 2 (Θ 1 ), • • • , σ 2 (Θ n )}. Consider that n -1 Θ T WΣWΘ = n -1 n i=1 {V C (Θ i -θ)} 2 σ 2 (Θ i ) n -1 n i=1 {V C (Θ i -θ)} 2 S T Θi-θ σ 2 (Θ i ) n -1 n i=1 {V C (Θ i -θ)} 2 S Θi-θ σ 2 (Θ i ) n -1 n i=1 {V C (Θ i -θ)} 2 S Θi-θ S T Θi-θ σ 2 (Θ i ) , (10)
and approximate the components of the above matrix using the following relationships

1 n n i=1 {V C (Θ i -θ)} 2 σ 2 (Θ i ) = [0,2π) d {V C (Θ i -θ)} 2 σ 2 (α) f (α)dα + o p (1) = I 0 (2κ) 2π{I 0 (κ)} 2 d σ 2 (θ) f (θ){1 + o p (1)} ; 1 n n i=1 {V C (Θ i -θ)} 2 S T Θi-θ σ 2 (Θ i ) = [0,2π) d {V C (α i -θ)} 2 S T α-θ σ 2 (α) f (α)dα + o p (1) = o p (1) ; 1 n n i=1 {V C (Θ i -θ)} 2 S Θi-θ S T Θi-θ σ 2 (Θ i ) = [0,2π) d {V C (α i -θ)} 2 S α-θ S T α-θ σ 2 (α) f (α)dα + o p (I) = F (2, κ 2 ) 4π{I 0 (κ)} 2 I 0 (2κ) 2π{I 0 (κ)} 2 d-1 σ 2 (θ) f (θ){I + o p (I)} , where F (2, κ 2 ) := {I 0 (k)} 2 + {I 1 (k)} 2 + 2 ∞ j=2 I j (κ){I j (κ) -I j-2 (κ)
} is the regularized confluent hypergeometric function of the first kind. Combining the previous results with the approximations in (9), and recalling assumption ii), we finally obtain

Var[ m(θ; C) | Θ 1 , • • • , Θ n ] = I 0 (2κ) 2π{I 0 (κ)} 2 d σ 2 (θ) n f (θ) + o p (n -1 κ d/2
) .

Proof of Corollary 1. Replace I 1 (κ)/I 0 (κ) by 1 with an error of magnitude O(κ -1 ), and use Proof of Theorem 4. Follow the proof of Theorem 4.1 of [START_REF] Ruppert | Multivariate locally weighted least squares regression[END_REF] with these two recommendations: in the design matrix replace (X ix) j , with sin j (Θ i -θ), and use the expansion f (u+θ) = f (θ)+sin(u) f ′ (θ)+O{sin 2 (u)}.

Figure 1 :

 1 Figure 1: Relative efficiency of Fejér (--), wrapped normal (-----), and von Mises (• • • ) kernels to the Dirchlet kernel, for various values of n.With respect to the underlying true density, the left group corresponds to the von Mises distribution with ρ = I 1 (ν)/I 0 (ν), while the right group corresponds to the wrapped Cauchy distribution.

Figure 2 :

 2 Figure 2: Comparison of averaged squared error as a function of κ over 200 simulations (dashed line), and asymptotic mean squared error given by Theorem 2 (continuous line) with locations of minima. Top row: n = 50; lower row: n = 500, with m estimated at θ = 0 (left), θ = 2 (middle) and θ = 3 (right).

Figure 3 :

 3 Figure 3: Comparison of averaged squared error as a function of κ over 200 simulations (dashed line), and asymptotic mean squared error given by Theorem 2 (continuous line) with locations of minima shown by the integers 2, 3, 4 which corresponds to the dimension of the data. m is estimated at θ = (0, • • • , 0) T (left) and θ = (π/2, • • • , π/2) T (right).

,

  then minimize the asymptotic MSE. Proof of Theorem 3. Follow the proof of Theorem 2, with K C (Θ iθ) as ith entry of the weight matrix, i = 1, • • • , n. In particular, to derive the conditional bias firstly note thatn -1 Θ T WΘ ⋍ f (θ) + o p (1) 1/2{1γ 2 (κ)}D T f (θ) + o p (1) 1/2{1γ 2 (κ)}D f (θ) + o p (1) 1/2{1γ 2 (κ)} f (θ)I + o p (I) ,and, in virtue of assumption i) of Theorem 1,e T 1 (n -1 Θ T WΘ) -1 ⋍ { f (θ)} -1 + o p (1) -D T f (θ){ f (θ)} -2 + o p (1) . γ 2 (κ)}tr{H m (θ)} f (θ) + o p (1) O p (1) , to get E[ m(θ; C)m(θ) | Θ 1 , • • • , Θ n ] = 1 4 {1γ 2 (κ)}tr{H m (θ)} + o p (1) .To derive the conditional variance, observe that the upper-left entry of the matrix (10) generalizes as1 n n i=1 {K C (Θ iθ)} 2 σ 2 (Θ i ) ⋍ R(K C )σ 2 (θ) f (θ){1 + o p (1)}, where R(K C ) = {R(K κ )} d = {(2π) -1 (1 + 2 ∞ j=1 γ 2 j (κ))} d, the diagonal blocks are o p (1), whereas lettingA(K C ) := [γ 2 0 (κ) + γ 2 1 (κ) + 2 ∞ j=2 γ j (κ){γ j (κ)γ j-2 (κ)}]{R(K κ )} d-1 4π , Θ iθ)} 2 S Θi-θ S T Θi-θ σ 2 (Θ i ) ⋍ A(K C )σ 2 (θ) f (θ){I + o p (I)} .Hence, it finally resultsVar[ m(θ; C) | Θ 1 , • • • , Θ n ] = R(K C )σ 2 (θ) n f (θ) {1 + o p (1)} .

  and Y i ∈ Ê are both observable, absolutely continuous, random variables taking values respectively in [0, 2π) d and Ê. ¿From now on we will assume that

  where Θ i s are i.i.d. observations from the circular design density f , and Y i s are i.i.d. real-valued random variables , take the local linear kernel regression estimator m(• ; C) equipped with the weight function V C

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Theorem 4. Given the dataset {(Θ i , Y i ), i = 1, • • • , n}, where Θ i s are i.i.d. observations from the circular onedimensional density f , and Y i s are i.i.d. real-valued random variables, take the local pth degree polynomial regression estimator m(• ; κ) equipped with a second-order circular kernel K κ . Assume conditions i) of Theorem 1, iii) and iv) of Theorem 2. Moreover, assume that i) for the kernel

Then, for any θ ∈ [0, 2π),

and

Proof. See Appendix.

Simulation results

We briefly explore the asymptotic result given by Theorem 2 in a simulation study. We first investigate the dependence of the mean squared error on θ, n and κ when d = 1 and choose a sharp-peaked response

, n coming from a von Mises density with mean π and concentration parameter 1. We estimate m(θ) at θ = 0, 2, 3 and compare the average squared error of (4) with the asymptotic mean squared error given in Theorem 2 over κ for n = 50 and n = 500. The results are displayed in Figure 2, and the asymptotic nature of the result is clear. Note that the values of the second derivative of m at θ = 0, 2, 3 are -0.59, 0.98, 140.89, respectively, which explains the poorer performance at θ = 3. Secondly, we explore the dependence on d. In this case we use the model

and f is a product of (independent) von Mises densities with mean zero and concentration parameter 1. We estimate m(θ) at θ = (0, • • • , 0) T and (π/2, • • • , π/2) T for a range of κ, for n = 500. Figure 3 shows good agreement for d = 2 between the average squared error and the asymptotic mean squared error. However, we note increasingly poor behaviour as d increases, indicating that the asymptotic nature of the result also depends on d, and again illustrating the well-known phenomenon of the curse of dimensionality.

Appendix

Proof of Theorem 1. Express K κ (θ) in terms of a Fourier series, and, recalling that for very small values of u sin(u) ≃ u, use the expansion

Then, starting from (2), make a change of variable and use assumption i) to get

A C C E P T E D M A N U S C R I P T

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In particular, to derive the conditional bias, let Q p be the matrix of order p + 1 having as (i, j) entry η i+ j-1 (K κ ), and observe that, in virtue of assumption

, with E p being the matrix defined in Remark 2, to get

where r 1 is a (p + 1) × 1 vector having 1 as first entry and 0 elsewhere. For the conditional variance, denoting as T p the matrix of order p + 1 having sin i+ j-2 (u){K κ (u)} 2 du as (i, j) entry, and recalling condition i), it follows that n -1 Θ T W 2 Θ = f (θ)T p + o p (I).

Aknowledgements

The authors are grateful to two anonymous referees for their valuable comments which led to improvements in this article.


Aniello Buonocore
email: aniello.buonocore@unina.it

Enrica Pirozzi
email: enrica.pirozzi@unina.it

Luigia Caputo
email: luigia.caputo@unito.it

A note on the sum of uniform random variables

Keywords: induction, recursive formulas 2000 MSC: 60G50, 60E99

An inductive procedure is used to obtain distributions and probability densities for the sum S n of independent, non equally uniform random variables. Some known results are then shown to follow immediately as special cases. Under the assumption of equally uniform random variables some new formulas are obtained for probabilities and means related to S n . Finally, some new recursive formulas involving distributions are derived.

Introduction

The problem of calculating the distribution of the sum S n of n uniform random variables has been the object of considerable attention even in recent times. The motivation can be ascribed to various reasons such as the necessity of handling data drawn from measurements characterized by different level of precision [START_REF] Bradley | On the distribution of the sum of n non-identically distributed uniform random variables[END_REF], or questions appearing in change point analysis [START_REF] Sadooghi-Alvandi | On the distribution of the sum of independent uniform random variables[END_REF], or, more in general, the need of aggregating scaled values with differing numbers of significant figures [START_REF] Potuschak | More on the distribution of the sum of uniform random variables[END_REF]). It appears that this problem has been taken up first in [START_REF] Olds | A note on the convolution of uniform distributions[END_REF], where by somewhat obscure procedures formulas for the probability density function of S n and its distribution function are derived. An accurate bibliography of articles published in the last century is found in [START_REF] Bradley | On the distribution of the sum of n non-identically distributed uniform random variables[END_REF], where the authors also obtain the probability density function of S n by non probabilistic arguments, namely via a complicated analytical inversion of the characteristic function. Such a procedure was successively and successfully simplified in [START_REF] Potuschak | More on the distribution of the sum of uniform random variables[END_REF], where again no trace of probabilistic arguments is present. An attempt to achieve the same results by a simpler procedure appears in [START_REF] Sadooghi-Alvandi | On the distribution of the sum of independent uniform random variables[END_REF] where a given function is assumed to be the unknown probability density function, the proof of the correctness of such an ansatz being that its Laplace transform coincides with the moment generating function of S n . Quite differently, the present note includes a novel proof of the above cited results (Proposition 2.1). This is based on an inductive procedure, suitably adapted to our general instance, used by [START_REF] Feller | An introduction to probability theory and its applications[END_REF] for the case of identically distributed variables, that further pinpoints the usefulness of induction procedures in the probability context. (See also [START_REF] Hardy | Inequalities[END_REF] for some more illuminating examples.) In the case of identically distributed random variables, some results concerning certain probabilities and means of random variables related to S n are obtained (Lemma 3.1, Theorem 3.1, corollaries 3.1 and 3.2, Proposition 3.4), as well as certain recurrence relations that are reminiscent of those holding for Stirling numbers (Propositions 3.5,3.6,3.7).

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The general case

Let {X n } n∈N denote a sequence of uniform distributed independent random variables and denote S n = n i=1 X i . Without loss of generality we assume that X n ∼ U (0, a n ) with a n positive real numbers. By adopting a suitably modified procedure due to [START_REF] Feller | An introduction to probability theory and its applications[END_REF] we shall obtain the probability density function f n (x) and the distribution function F n (x) of S n for all n ∈ N. The starting point is to write

F Xn (x) = x + -(x -a n ) + a n , ∀n ∈ N, ∀x ∈ R, (1)
where

(x -c) + = max{x -c, 0}, ∀c ∈ R. Next we shall make use of x -∞ (y -c) + n-1 dy = 1 n (x -c) + n , ∀n ∈ N, ∀c ∈ R + . (2)
In addition we note that, by convolution, probability density functions and distribution functions are related as follows:

f n+1 (x) = an+1 0 f n (x -y)f Xn+1 (y) dy = F n (x) -F n (x -a n+1 ) a n+1 , ∀n ∈ N, ∀x ∈ R. (3)
Claim 2.1. One has

F 1 (x) = x + -(x -a 1 ) + a 1 , ∀x ∈ R (4)
and

f 2 (x) = x + -(x -a 1 ) + -(x -a 2 ) + + [x -(a 1 + a 2 )] + a 1 a 2 , ∀x ∈ R. (5)
Proof. It follows from (1) written for S 1 ≡ X 1 , and from (3). 2

Claim 2.2. One has

F 2 (x) = (x + ) 2 -[(x -a 1 ) + ] 2 -[(x -a 2 ) + ] 2 + {[x -(a 1 + a 2 )] + } 2 2a 1 a 2 , ∀x ∈ R (6)
and

f 3 (x) = (x + ) 2 -(x -a 1 ) + 2 -(x -a 2 ) + 2 -(x -a 3 ) + 2 + + [x -(a 1 + a 2 )] + 2 + [x -(a 1 + a 3 )] + 2 + [x -(a 2 + a 3 )] + 2 (7) - [x -(a 1 + a 2 + a 3 )] + 2 (2a 1 a 2 a 3 ) -1 , ∀x ∈ R.
Proof. Eq. ( 6) follows from ( 5) and (2). From ( 6) and (3) one then obtains Eq. ( 7). 2

Claims 2.1 and 2.2 lead us to infer a possible general forms of the distribution function of S n and of the probability density function of S n+1 , as specified in the following Proposition.

Proposition 2.1. The distribution function F n (x) of S n and the probability density function f n+1 (x) of S n+1 are given by, respectively:

F n (x) = 1 n! A n (x + ) n + n ν=1 (-1) ν n j1=1 n j2=j1+1 • • • n jν =jν-1+1 [x -(a j1 + a j2 + • • • + a jν )] + n , ∀n ∈ N, ∀x ∈ R (8)

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and

f n+1 (x) = 1 n! A n+1 (x + ) n + n+1 ν=1 (-1) ν n+1 j1=1 n+1 j2=j1+1 • • • n+1 jν =jν-1+1 [x -(a j1 + a j2 + • • • + a jν )] + n , ∀n ∈ N, ∀x ∈ R. (9)
Proof. We proceed by induction. Claims 2.1 and 2.2 show that Eqs. ( 8) and ( 9) hold for n = 1 and n = 2.

Let us now assume that they hold for n = r -1 and prove that they also hold for n = r. To this purpose, we re-write Eq. ( 9) for n = r -1 and x = y and then integrate both sides over (-∞, x). By virtue of (2), Eq. ( 8) with n = r then follows. To obtain Eq. ( 9) for n = r we make use of (3) and of the just obtained expression of F r (x). Hence,

f r+1 (x) = 1 r! A r+1 (x + ) r + r ν=1 (-1) ν r j1=1 r j2=j1+1 • • • r jν =jν-1+1 [x -(a j1 + a j2 + • • • + a jν )] + r + -(x -a r+1 ) + r + (10) - r ν=1 (-1) ν r j1=1 r j2=j1+1 • • • r jν =jν-1+1 [x -(a j1 + a j2 + • • • + a jν + a r+1 )] + r
.

Eq. ( 10) identifies with Eq. ( 9) written for n = r since the curly brackets contains all and only all the following terms:

1. [(x) + ] r ; 2. [(x -a 1 ) + ] r , [(x -a 2 ) + ] r , . . . , [(x -a r+1 ) + ] r ; 3. for 1 < ν ≤ r (-1) ν r j1=1 r j2=j1+1 • • • r jν =jν-1+1 [x -(a j1 + a j2 + • • • + a jν )] + r + -(-1) ν-1 r j1=1 r j2=j1+1 • • • r jν-1=jν-2+1 x -(a j1 + a j2 + • • • + a jν-1 + a r+1 ) + r ≡ (-1) ν r+1 j1=1 r+1 j2=j1+1 • • • r+1 jν =jν-1+1 [x -(a j1 + a j2 + • • • + a jν )] + r ; 4. (-1) r+1 [x -(a 1 + a 2 + • • • + a r+1 ] + r .
This complete the induction. 2

A special case

Let us assume that the random variables in {X n } n∈N are identically distributed.

Proposition 3.1. When a n = a > 0 for all n ∈ N then

F n (x) = 1 n! a n n ν=0 (-1) ν n ν (x -νa) + n , ∀n ∈ N, ∀x ∈ R (11)
and

f n+1 (x) = 1 n! a n+1 n+1 ν=0 (-1) ν n + 1 ν (x -νa) + n , ∀n ∈ N, ∀x ∈ R. (12)
Hereafter, for simplicity we shall take a n = a = 1 for all n ∈ N. Then, from Eqs. (3) there follows

f n+1 (x) = F n (x) -F n (x -1), ∀n ∈ N, ∀x ∈ R (13) so that f n+1 (k) = F n (k) -F n (k -1), ∀n ∈ N, k ∈ {0, 1, . . . , n + 1}, ( 14
)
whereas from Eqs. ( 11) and ( 12) one obtains

F n (k) = 1 n! k ν=0 (-1) ν n ν (k -ν) + n , ∀n ∈ N, k ∈ {0, 1, . . . , n} (15)
and

f n (k) = 1 (n -1)! k ν=0 (-1) ν n ν (k -ν) + n-1 , ∀n ∈ N, k ∈ {0, 1, . . . , n}. ( 16
)
Proposition 3.2. When a n = a = 1 for all n ∈ N then

F n (x) = k j=1 f n+1 (x + j -k), ∀n ∈ N, k ∈ {1, 2, . . . , n}, k -1 ≤ x ≤ k. ( 17
)
Proof. Starting from ( 13), by iteration it follows that

F n (x) = f n+1 (x) + F n (x -1) = f n+1 (x) + f n+1 (x -1) + F n (x -2) = • • • = k j=1 f n+1 (x + j -k) + F n (x -k). Since x -k ≤ 0, one has F n (x -k) = 0, which completes the proof. 2 47 Proposition 3.3. When a n = a = 1 for all n ∈ N then k k-1 F n (x) dx = F n+1 (k), ∀n ∈ N, k ∈ {1, 2, . . . , n}. (18)

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Proof. Making use of ( 17) one obtains

k k-1 F n (x) dx = k j=1 k k-1 f n+1 (x + j -k) dx = k j=1 F n+1 (x + j -k) k k-1 = k j=1 [F n+1 (j) -F n+1 (j -1)] = F n+1 (k) -F n+1 (0).
The proof is then a consequence of F n (0) = 0 for all n ∈ N. 2

Consider now the event S n,k = {k -1 ≤ S n ≤ k} and let P n,k := P (S n,k ). From ( 14) it follows that

P n,k = F n (k) -F n (k -1) = f n+1 (k), ∀n ∈ N, k ∈ {1, 2, . . . , n}. (19)
Lemma 3.1. When a n = a = 1 for all n ∈ N then

P (S n+1 ≤ k, S n,k ) = F n+1 (k) -F n (k -1), ∀n ∈ N, k ∈ {1, 2, . . . , n}. (20)
Proof. Let n ∈ N and 1 ≤ k ≤ n. Then,

P (S n+1 ≤ k, S n,k ) = P (X n+1 ≤ k -S n , S n,k ) = T f Xn+1 (x)f n (y) dx dy
where T denotes the domain in the x-y plane defined by 0 < x < 1 and k -1 < y < kx. Hence, by integration along the y-axis from k -1 to kx, for all x ∈ (0, 1) we obtain

P (S n+1 ≤ k, S n,k ) = 1 0 dx k-x k-1 f n (y) dy = 1 0 F n (k -x) dx -F n (k -1) = k k-1 F n (x) dx -F n (k -1). ( 21
)
Eq. ( 20) follows from ( 21) and ( 18). 2

Lemma 3.1 will be used to prove the following theorem.

Theorem 3.1. When a n = a = 1 for all n ∈ N then

P (S n+1 ≤ k|S n,k ) = k n + 1 , ∀n ∈ N, k ∈ {1, 2, . . . , n}. (22)
Proof. Let k = 1. ∀n ∈ N, from (15) there follows F n (1) = 1/n!, whereas F n (0) = 0. Hence, making use of ( 19) one obtains

P (S n+1 ≤ 1|S n,1 ) = P (S n+1 ≤ 1, S n,1 ) P n,1 = F n+1 (1) -F n (0) F n (1) -F n (0) = n! (n + 1)! = 1 n + 1 . ( 23
)
This proves ( 22) for k = 1. From (23) it follows that

1 n + 1 ≡ P (S n+1 ≤ 1|S n,1 ) = E [P (X n+1 ≤ 1 -y|S n,1 , S n = y)] = 1 -E [S n |S n,1 ] which ultimately implies E [S n |S n,1 ] = n n + 1 .
Since X 1 , X 2 , . . . , X n are uniform iid random variables, the mean of each of them conditional on S n,1 is 1/(n + 1). Hence, given that S n,1 occurs, the means of S 1 , S 2 , . . . , S n partition [0, 1] into n + 1 equally wide intervals. Therefore, for 1 < k ≤ n, if S n,k occurs, the interval that is partitioned into n + 1 equally wide intervals is now [0, k]. This implies that X n+1 cannot exceed k/(n + 1) to insure that S n+1 remains below k. 2

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Corollary 3.1. When a n = a = 1 for all n ∈ N then

E [S n |S n,k ] = n n + 1 k, ∀n ∈ N, k ∈ {1, 2, . . . , n}. (24)
Proof. Due to Theorem 3.1 one has

k n + 1 ≡ P (S n+1 ≤ k|S n,k ) = E [P (X n+1 ≤ k -y|S n,k , S n = y)] = k -E [S n |S n,k ]
which ultimately yields Eq. ( 24). 2

Corollary 3.2. When a n = a = 1 for all n ∈ N then Eq. ( 26) finally follows by equating the right hand sides of ( 27) and (28). 2

E S n 1 S n,k = n n + 1 kP n,k , ∀n ∈ N, k ∈ {1, 2, . . . , n}. (25
The forthcoming recursive formulas are a consequence of Theorem 3.1.

Proposition 3.5. When a n = a = 1 for all n ∈ N then

F n+1 (k) = F n (k) k n + 1 + F n (k -1) n + 1 -k n + 1 , ∀n ∈ N, k ∈ {1

F

  When a n = a = 1 for all n ∈ N then n n+1 (k), ∀n ∈ N, k ∈ {1, 2, . . . , n}.(26)Proof. Let n ∈ N. From (19) and (25) we obtain

Note that Eqs. (11) and (12) obtained by us as a special case of (8) and (9) are in agreement with a result due to[START_REF] Feller | An introduction to probability theory and its applications[END_REF].

Acknowledgements

We wish to thank Professors R. Johnson and L.M. Ricciardi for helpful comments.

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Proof. Eq. ( 11) follows from (8) after noting that now A n = a n and that a j1 + a j2 + • • • + a jν = νa for ν = 0, 1, . . . , n. Indeed, in the sum on ν in (8), the term in curly bracket becomes [(xνa) + ] n , so that

Eq. ( 12) follows from (9) by a similar argument. 1 2

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Proof. Let n ∈ N and 1 ≤ k ≤ n + 1. From ( 19) and (20) one obtains

On the other hand,

Hence, from ( 19) and ( 22) one derives

Eq. ( 29) then immediately follows after equating the right hand sides of ( 30) and ( 31). 2

Note that (29) trivially holds also for k = 0, yielding 0 = 0.

Eq. ( 25) can be alternatively obtained via (29).

Proposition 3.6. When a n = a = 1 for all n ∈ N then

Proof. Let n ∈ N and 1 ≤ k ≤ n + 1. By difference of Eqs. (29) written for k and for k -1, one obtains

Proof. Let n ∈ N and 1 ≤ k ≤ n + 1. From ( 19) and (32) it follows that

By making again use of (19), Eq. ( 33) is finally obtained. 2