David Blondin
email: blondin@ccr.jussieu.fr

Rates of strong uniform consistency for local least squares kernel regression estimators

Keywords: nonparametric regression, derivative estimation, kernel estimation, local linear least squares kernel estimator, local polynomial fitting, strong uniform consistency, rate of convergence, uniform limit law of the logarithm

We establish exact rates of strong uniform consistency for the multivariate Nadaraya-Watson kernel estimator of the regression function and its derivatives. As a special case, we treat the local linear estimator of the regression and the local polynomial smoothers of derivatives of the regression in the more convenient univariate setting. Our methods of proofs are based upon modern empirical process theory in the spirit of the results of Einmahl and Mason [4] and Deheuvels and Mason [2] relative to uniform deviations of nonparametric kernel estimators.

Introduction

Let (X, Y) = (X 1 , Y 1 ), (X 2 , Y 2 ), . . ., be independent and identically distributed random vectors in IR p ×IR d . In this paper, we are concerned with the estimation of the conditional expectation of a functional ψ of Y given X. The following notation and assumptions will be in force. Let x (respectively y) denote a vector in IR p (resp. IR d ). We assume that the distribution function of (X, Y) has a Lebesgue density f X,Y (x, y) = g(x, y) on IR p × IR d . We denote by f X (x) = f (x) the marginal density of X assumed to exist for each x ∈ IR p . Let ψ : IR d → IR q a Borel measurable function, bounded on each compact set of IR d . We consider the regression function of ψ(Y) given X = x, defined by

m ψ (x) = IE ψ(Y)|X = x = 1 f(x) IR d ψ(y)g(x, y)dy := r ψ (x) f(x) , (1.1)
whenever this regression function is meaningful (see, e.g., (F.1-2) below). Throughout this paper, we will investigate the strong consistency of kernel-type estimators of the regression m ψ (x) and its derivatives. We will work in the multivariate framework, where p ≥ 1, q ≥ 1, d ≥ 1 are arbitrary integers, and impose the following set of assumptions upon the distribution of (X, Y).

(F.1) g(•, •) is continuous on J × IR d and f(•) is continuous and strictly positive on J;

(F.2) Y1I {X∈J} is bounded on IR d .

Here and elsewhere k ≥ 1 denotes a fixed order of differentiation in the following sense. Let ζ = (ζ 1 , . . . , ζ q ) be an arbitrary measurable function with ζ j : IR p → IR, j = 1, . . . , q. In this section, we consider estimation of functionals of ζ defined at x = (x 1 , . . . , x p ) ∈ J. For each p-uple of non-negative integers

k 1 ≥ 0, . . . , k p ≥ 0, k = (k 1 , . . . , k p ), we introduce the differential operator D (k) of order |k| = k = k 1 + . . . + k p , defined by D (k) ζ(x) = ζ (k) (x) = D (k) ζ 1 (x), . . . D (k) ζ q (x) ,
where

D (k) ζ j (x) = ∂ ∂x 1 k1 . . . ∂ ∂x p kp ζ j (x), j = 1, . . . , q.
At first, we treat the Nadaraya-Watson kernel estimator (see [START_REF] Nadaraya | On estimating regression[END_REF] and [START_REF] Watson | Smooth regression analysis[END_REF])) and its partial derivatives when the predictor variables are IR p -valued. For this purpose we need to introduce a general kernel function K : IR p → IR, fulfilling the conditions

(K.1) (i) K is bounded; (ii) K(x 1 , . . . , x p ) is a right-continuous function of x 1 , . . . , x p ;
(iii) K(u) = Φ(P (u)), where P is a polynomial, and Φ is a real valued function of bounded variation;

(K.2) K is compactly supported; (K.3) IR p K(u) du = 1;

(K.4) K is k-times differentiable, with partial derivatives satisfying (K.1).

Let h n , n = 1, 2 . . ., denotes a non-random (bandwidth) sequence of positive constants satisfying the following assumptions: as n → ∞,

(H.1) h n 0 and nh n ∞, (H.2) nh 2k+p n / log(h -p n ) → ∞, (H.3) log(h -p n )/ log log n → ∞.
We introduce the kernel estimators of f (x), r ψ (x) and m ψ (x), defined by

fn (x) = 1 nh p n n i=1 K x -X i h n , rψ;n (x) = 1 nh p n n i=1 ψ(Y i )K
x -X i h n and mψ;n (x) = rψ;n (x) fn (x) when fn (x) = 0.

Our estimators of the functional D (k) of f(x) and r ψ (x) are then defined as follows

f (k) n (x) = D (k) fn (x) = 1 nh k+p n n i=1 K (k) x -X i h n ,
A c c e p t e d m a n u s c r i p t

r(k) ψ;n (x) = D (k) rψ;n (x) = 1 nh k+p n n i=1 ψ(Y i )K (k) x -X i h n .
Unless otherwise specified, we will limit most of our exposition to the case where k = (k 1 , . . . , k p ) is such that k j = 1 and k l = 0 for l = j and denote by k j = (0, . . . , 1, . . . , 0) the corresponding p-uple. It will become obvious later on that our methods allow to treat likewise the case of an arbitrary k. Thus m(kj)

ψ;n (x) = D (kj ) mψ;n (x) = r(kj) ψ;n (x) fn (x) - rψ;n (x) f (kj ) n (x) f 2 n (x)
. We note that, when k j ≥ 2 for some j, m(k) ψ;n (x) = D (k) mψ;n (x) may be obtained likewise through the usual Leibniz expansion of derivatives of products. Moreover, one of the motivation to this study comes from the fact that the estimation of the proper partial derivatives of m ψ is needed to implement plug-in bandwidth selection strategies (which is the crucial part in constructing efficient kernel estimators).

It will be convenient to center mn (x) and m(kj) ψ;n (x), respectively, by

IE mψ;n (x) := IE r(k) ψ;n (x) IE f (k) n (x) =: r ψ;n (x) f n (x) , IE m(kj) ψ;n (x) := r (kj ) ψ;n (x) f n (x) - r ψ;n (x)f (kj ) n (x) f 2 n (x)
.

(1.2)

A similar setup applies for operators

D (k) with |k| = k ≥ 2.
The main purpose of the present paper is to give a description of the exact rate of strong convergence of the uniform deviation of m(k) ψ;n (x)-IE m(k) ψ;n (x) over x ∈ I. The convergence rate we shall obtain takes the form of uniform law of the logarithm and are stated in Section 2. The methods of proof that we will use to establish these results can be applied to treat a number of other kernel-type estimators, including local polynomial estimators (see, e.g., Fan and Gijbels [START_REF] Fan | Local Polynomial Modelling and Its Applications[END_REF] and Ruppert and Wand [START_REF] Wand | Kernel Smoothing[END_REF]). Section 3 contains the one-dimensional extension to the local linear least squares kernel estimator of m ψ , higher-order polynomial fits and derivative estimation. Section 4 is devoted to the proofs. The main idea underlying our arguments is that by using the techniques from abstract empirical process theory and probability on Banach spaces, one can prove almost sure limit laws for a large class of nonparametric estimators of regression functions.

Uniform laws of the logarithm for estimators of the regression derivatives

First, we treat the special case q = 1 (that is, when ψ(Y) ∈ IR).

Theorem 2.1 Under (F.1-2), (H.1-3), (K.1-4) we have, as n → ∞, nh 2k+p n 2 log(h -p n ) 1/2 sup x∈I ± m(k) ψ;n (x) -IE m(k) ψ;n (x) -σ ψ (I) = o(1), almost surely, (2.1)
where

σ ψ (I) = sup x∈I Var ψ(Y) X = x f(x) IR p [K (k) (u)] 2 du 1/2 . (2.2)
For a related 'in probability' theorem, refer to Theorem 1.1 in [START_REF] Deheuvels | General asymptotic confidence bands based on kernel-type function estimators[END_REF].

The strictly multivariate case:

ψ(Y) ∈ IR q , q > 1
The extension to several dimensions for ψ(Y) is somewhat straightforward. We just need a suitable preliminary normalization of the stochastic deviation m(k)

ψ;n (x) -IE m(k) ψ;n (x)
. This enables us to characterize precisely the corresponding limit set, by using a standard argument developped by Finkelstein

A c c e p t e d m

a n u s c r i p t [START_REF] Finkelstein | The law of the iterated logarithm for empirical distributions[END_REF]. Let Σ ψ (x) denotes the conditional variance-covariance matrix of ψ(Y) given that X = x, which is assumed definite positive without loss of generality. We will make use of the following asymptotic variance-covariance matrix

V x = V ψ (x) := 1 f (x) IR p [K (k) (t)] 2 dt × Σ ψ (x),
which is properly defined for x ∈ J and definite positive under (F.1-3). Below, suprema of q-vectors v = {v 1 , . . . , v q } T are meant with respect to the maximum norm | • | + on IR q , where |v| + = max

i≤q |v i |.
Theorem 2.2 If (F.1-2), (H.1-3) and (K.1-4) hold, then, the sequence of normalized q-vectors

nh 2k+p n 2 log(h -p n ) 1/2 sup x∈I ± V -1/2 x m(k) ψ;n (x) -IE m(k) ψ;n (x)
is almost surely relatively compact in IR q with limiting set

S q = v ∈ IR q : v T v = 1
, the q-dimensional unit sphere.

Notice that Theorem 2.2 is a simple consequence of Theorem 2.1, when combined with the proof of Lemma 2 of Finkelstein (1971) [START_REF] Finkelstein | The law of the iterated logarithm for empirical distributions[END_REF]. To see how Theorem 2.2 follows from Theorem 2.1, observe that the normalized deviation (properly rescaled)

V -1/2 x m(k) ψ;n (x) -IE m(k) ψ;n (x)
is asymptotically normally distributed with an identity covariance matrix.

Uniform law of the logarithm for local polynomial kernel estimators

In this paper, we are mainly focused on the stochastic part of the usual deviation. In the literature, it is a well-known fact that the asymptotic bias of the Nadaraya-Watson estimator has a bad form. To overcome this problem, it exists many alternative estimators. For example, if we assume some regularity conditions on the regression function, one can use the local polynomial regression techniques of Fan and Gijbels [START_REF] Fan | Local Polynomial Modelling and Its Applications[END_REF].

Our next task will be to extend the preceding results to the local polynomial least squares smoothers. For ease of presentation, we restrict ourselves first to the univariate case and to the local linear least squares estimator. At the end of this section, we present also some extensions to higher-order polynomial fits and derivative estimation.

Let (X, Y ), (X 1 , Y 1 ), (X 2 , Y 2 ), . . ., be independent and identically distributed random couples in IR × IR. (K.2) K is compactly supported and IR K(u) du = 1;

(H.1)

h n 0 and nh n ∞, (H.2) nh 2k+1 n / log(h -1 n ) → ∞, (H.3) log(h -1 n )/ log 2 n → ∞, as n → ∞.

A c c e p t e d m a n u s c r i p t

For a kernel K fulfilling (K.1-2), we set for any integer r ≥ 0,

µ r = µ r (K) = IR u r K(u)du.
It is common to work with a kernel function K(•) such that (K.3) µ 1 (K) = 0 and µ 2 (K) = 0.

Our first aim will be to establish the strong uniform consistency of the local linear estimator of the regression, defined by mLL ψ;n (x) := rψ;n (x) fn,2 (x) -rψ;n,1 (x) fn,1 (x) fn (x) fn,2 (x) -fn,1 (x) fn,1 (x) ,

where, for j = 1, 2, fn,j (x

) := 1 nh n n i=1 x -X i h n j K x -X i h n and rψ;n,1 (x) := 1 nh n n i=1 ψ(Y i ) x -X i h n K x -X i h n .
Notice that, in the multivariate setting, such simple explicit formula for the local linear estimator is not available. This estimator is better than the Nadaraya-Watson estimator when the design is random and has the favorable property to reproduce polynomial of degree one. In particular, Fan [START_REF] Fan | Design-adaptative nonparametric regression[END_REF] shows that the local linear estimator has an important asymptotic minimax property. Precisely, the local linear estimator has a high minimax efficiency among all possible estimators, including nonlinear smoothers such as median regression. For the centering terms, we proceed as in (1.2) and set

IE mLL ψ;n (x) := r ψ;n (x)f n,2 (x) -r ψ;n,1 (x)f n,1 (x) f n (x)f n,2 (x) -f n,1 (x)f n,1 (x)
.

We obtain the following uniform law of the logarithm concerning the local linear smoother.

Theorem 3.1 Under (F.1-2), (H.1-3), (K.1-3) we have, as n → ∞, We investigate now general polynomial fits and derivative estimation. Let p be the degree of the local polynomial fit. Throughout the end of this section, we will therefore assume that m ψ (•) is p times differentiable over the compact interval J and (p + 1) times differentiable over the compact interval I.

nh n 2 log(h -1 n ) 1/2 sup x∈I ± mLL ψ;n (x) -IE mLL ψ;n (x) -σ ψ (I) = o(
Set

X x =      1 (X 1 -x) . . . (X 1 -x) p . . . . . . . . . 1 (X n -x) . . . (X n -x) p      , W x = diag K X i -x h n and Ψ(y) =      ψ(Y 1 ) . . . ψ(Y n )      .
According to the definition of Ruppert and Wand [START_REF] Ruppert | Multivariate weighted least squares regression[END_REF], for k ≤ p , the local polynomial kernel estimator of the k-th derivative of m ψ (x) is defined by,

m(k) ψ (x) = m(k) ψ;n (x; p) := k! e T k+1 (X T x W x X x ) -1 X T x W x Ψ(y),

A c c e p t e d m a n u s c r i p t

where e k+1 is the (p + 1) × 1 vector having 1 in the (k + 1)-th entry and zeros elsewhere. It is now convenient to set, for each j ∈ IN,

fj (x) = 1 nh n n i=1 X i -x h n j K X i -x h n and rj (x) = 1 nh n n i=1 ψ(Y i ) X i -x h n j K X i -x h n . Observe that m(k) ψ (x) = k! e T k+1 ×      f0 (x) f1 (x) . . . fp (x) . . . . . . . . . fp (x) fp+1 (x) . . . f2p (x)      -1 ×         r0 (x) r1 (x) . . . rp (x)         .
For example, if (k, p) = (0, 1), we get clearly the local linear estimator defined in (3.1), namely

m(0) ψ;n (x; 1) = (1, 0) ×   f0 (x) f1 (x) f1 (x) f2 (x)   -1 ×   r0 (x) r1 (x)   = r0 (x) f2 (x) -r1 (x) f1 (x) f0 (x) f2 (x) -f1 (x) f1 (x) .
Explicit formulae of local polynomial estimators became large as p grows due to matrix complexity. When (k, p) = (1, 2), we get that m(1)

ψ (x; 2) = r(x)( f2 (x) f3 (x) -f1 (x) f4 (x)) + r1 (x)( f (x) f4 (x) -f 2 2 (x)) + r2 (x)( f1 (x) f2 (x) -f (x) f3 (x)) f (x) f2 (x) f4 (x) + 2 f1 (x) f2 (x) f3 (x) -f 3 2 (x) -f (x) f 2 3 (x) -f 2 1 (x) f4 (x)
.

Let N p be the (p + 1) × (p + 1) matrix having (i, j)th entry equal to µ i+j-2 (K), 1 ≤ i, j ≤ p + 1 and let M k,p (u) be the same as N p , but with the (k+1)-th column replaced by (1, u, . . . , u p ) T . This notation allows us to introduce the kernel corresponding to the estimation of the kth derivative with local polynomial fitting of degree p (see, e.g., [START_REF] Fan | Local Polynomial Modelling and Its Applications[END_REF] or [START_REF] Wand | Kernel Smoothing[END_REF] pp. 135-137):

K (k,p) (u) := k! × |M k,p (u)| |N p | × K(u).
For (k, p) = (1, 2) and (k, p) = (2, 3), we obtain

|M 1,2 (u)| |N 3 | = u µ 2 (K) and |M 2,3 (u)| |N 3 | = u 2 -µ 2 (K) µ 4 (K) -µ 2 2 (K)
.

Thus, under (K.2-3),

K (1,2) (u) := uK(u) µ 2 (K) and K (2,3) (u) := 2 × u 2 -µ 2 (K) K(u) µ 4 (K) -µ 2 2 (
K) are kernels of order (1, 3) and (2, 4) respectively. More generally, it is easily established that K (k,p) satisfies

IR u j K (k,p) (u)du =      0, 0 ≤ j ≤ p, j = q k!, j = q β k,p = 0, j = p + 1.
Therefore (-1) k K (k,p) is an order (k, p + 1) kernel as defined by Gasser et al. [START_REF] Gasser | Kernels for nonparametric curve estimation[END_REF]. Such kernels are tailored for estimating derivatives of functions such as regression or density functions. Following Ruppert and Wand [START_REF] Ruppert | Multivariate weighted least squares regression[END_REF] or Wand and Jones [START_REF] Wand | Kernel Smoothing[END_REF], some routine analysis gives us the following important fact

Var m(k) ψ;n (x) = 1 nh 2k+1 n × Var ψ(Y ) X = x f (x) IR K (k,p) (u) 2 du (1 + o IP (1)).

A c c e p t e d m a n u s c r i p t

By repeating the same arguments as in the proof of Theorem 3.1, we can formulate a uniform limit law for each couple (k, p).

Theorem 3.2 Under (F.1-2), (H.1-3), (K.1-3) we have, as n → ∞,

nh 2k+1 n 2 log(h -1 n ) 1/2 sup x∈I ± m(k) ψ;n (x) -IE m(k) ψ;n (x) -σ (k,p) (I) = o(1), almost surely,
where

σ (k,p) (I) = sup x∈I Var ψ(Y ) X = x f (x) IR K (k,p) (u) 2 du 1/2 .

Proofs

The proofs of our theorems follow the same line and can be inferred with little effort from the proof of the Theorem 1 in Einmahl and Mason [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF]. For ease of conciseness, we decide only to prove Theorem 3.1. In order to study the asymptotic behavior of the local linear estimator we introduce a general local empirical process. For any j ∈ IN and continuous real valued functions c(•) and d(•) on the compact interval J, set for x ∈ J,

W n,j (x, ψ) = n i=1 c(x)ψ(Y i ) + d(x) K j x -X i h n -nIE c(x)ψ(Y ) + d(x) K j x -X h n , (4.1)
where

K j (u) = u j K(u), u ∈ IR. Notice that W n,j (x, ψ) = √ n α n (ϑ x ),
where α n denotes the bivariate empirical process based upon (X 1 , Y 1 ), . . . (X n , Y n ) and indexed by the class of functions

ϑ x (u, v) := c(x)ψ(v) + d(x) K j x -u h n , x ∈ I.
Theorem 4.1 Under (F.1-2), (H.1-3) and (K.1-2), as n → ∞, with probability one,

2nh n log(h -1 n ) -1/2 sup x∈I ± W n,j (x, ψ) -σ W (I) = o(1)
,

where σ 2 W (I) = sup x∈I IE c(x)ψ(Y ) + d(x) 2 X = x f (x) IR K j (u) 2 du.
Proof. The methodology of proof is exactly the same as in [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF], Theorem 1, p .4.

To obtain the upper bound, the authors use a maximal form of the Bernstein's inequality to estimate the supremum of the deviation on the nodes of a discrete grid and then use the Talagrand's inequality combined with a moment bound to control the difference between the original empirical processes and its values over the nodes of the grid. The main argument of the proof is based on Talagrand's exponential equality [START_REF] Talagrand | Sharper bounds for Gaussian and emprical processes[END_REF] for general empirical processes combined with an useful moment inequality for empirical processes indexed by classes of functions of Vapnik-Cervonenkis type (see, e.g., [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF], [START_REF] Einmahl | Uniform in bandwidth consistency of kernel-type functions estimators[END_REF], [START_REF] Giné | On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptative intervals[END_REF] and [START_REF] Giné | Rates on strong uniform consistency for multivariate kernel density estimators[END_REF]).

Regarding the inner bound part, they apply poissonization techniques in the border of the limiting set σ 2 W (I). In our case, we notice that the function K j (u) = u j [K(u)], j ≥ 0, is clearly a function with bounded variation on IR. Thus K j fulfills also (K. [START_REF] Collomb | Conditions nécessaires et suffisantes de convergence uniforme d'un estimateur de la régression, estimation des dérivées de la regression[END_REF][START_REF] Deheuvels | General asymptotic confidence bands based on kernel-type function estimators[END_REF]. By following the steps of their Theorem 1 and by replacing K by K j , we obtain easily Theorem 4.1. Details are omitted.

2

A c c e p t e d m a n u s c r i p t

The deviation mLL ψ;n (x) -IE mLL ψ;n (x) cannot be expressed as a linear functional of the bivariate empirical process. However we are going to show that the stochastic deviation behaves asymptotically like its linearized version which corresponds to a special case of the general process in (4.1). Set

n := sup x∈I mLL ψ;n (x) -IE mLL ψ;n (x) - 1 f (x) rψ;n (x) -r ψ;n (x) -m ψ (x) fn (x) -f n (x) .
Notice that the right-hand side above is exactly the process W n,j (x, ψ) defined in (4.1), where we have chosen c(x) = 1/f (x) and d(x) = -m ψ (x)/f (x) which are uniformly continuous on J via (F.1-2) in conjunction with Scheffé's lemma. The proof of Theorem 3.1 boils down to the following lemma:

Lemma 1 Under the assumptions of Theorem 3.1, we have, with probability one,

n = o nh n 2 log(h -1 n ) -1/2
.

Proof. Applying Theorem 4.1 with c(x) = 0, d(x) = 1, j = 0, 1, 2, and then with c(x) = 1, d(x) = 0 and j = 0, 1, we get that

nh n 2 log(h -1 n ) 1/2 sup x∈I ± fn (x) -f n (x)
a.s.

= O(1), (4.2)

nh n 2 log(h -1 n ) 1/2 sup x∈I ± fn,1 (x) -f n,1 (x)
a.s.

= O(1), (4.3)

nh n 2 log(h -1 n ) 1/2 sup x∈I ± fn,2 (x) -f n,2 (x) a.s.
= O(1),

A c c e p t e d m a n u s c r i p t

  Denote by I= p i=1 [a i , b i ] and J = p i=1 [a i , b i ] ⊃ I two fixed closed hyper-rectangles of IR p such that -∞ < a i < a i < b i < b i < ∞, for i = 1, . . . , p.

  Now K denotes a real valued kernel function defined on IR, ψ is a Borel function bounded on each compact subinterval of IR and I = [a, b], J = [a , b ] ⊃ I are two fixed compact intervals in IR. In the univariate setting, assumptions (F.1-2), (K.1-3) and (H.1-3) reduce to the followings: (F.1) g(•, •) is continuous on J × IR and f(•) is continuous and strictly positive on J; (F.2) Y 1I {X∈J} is bounded on IR; (K.1) K is a right-continuous function with bounded variation on IR;

2 . 3 . 1

 231 Remark By setting ψ(y) = 1I {y≤t} in (3.1), we obtain a new estimator of the conditional distribution function which shares the same interesting bias properties as the local linear estimator.

±± 2 n, 1 - r n f n, 2 -r n,1 f n,1 f n f n,2 -f 2 n, 1 = rn fn, 2 - 2 n, 1 + r n f n, 2 - 2 n, 1 fn fn,2 -f 2 n, 1 × f n f n,2 -f 2 2 × 1 A c c e p t e d m a n u s c r i p t + r n fn

 2122122122121221 rψ;n (x) -r ψ;n (x) rψ;n,1 (x) -r ψ;n,1 (x) .1-3), (K.1-3) and (H.1), we obtain, via Bochner's Lemma,f n (x) = f (x) + o(1),(4.7)f n,1 (x) = f (x)µ 1 (K) + o(1) = o(1),(4.8)f n,2 (x) = f (x)µ 2 (K) + o(1), r ψ;n (x) = r ψ (x) + o(1)),(4.9)r ψ;n,1 (x) = r ψ (x)µ 1 (K) + o(1) = o(1).(4.10)We leave temporarily the dependence on x and ψ,mLL n -IE mLL n = rn fn,2 -rn,1 fn,1 fn fn,2 -f r n f n,2 + r n,1 f n,1 -rn,1 fn,1 fn fn,2 -f r n,1 f n,1 f n f n,2 -f 2 n,1 -fn fn,2 + f rn -r n fn,2 + r n fn,2 -f n,2 + r n,1 -rn,1 f n,1 + rn,1 f n,1 -fn,fn,2 f n × f n -fn fn,2 + f n f n,2 -fn,2 + fn,1 -f n,1 fn,1 + f n,1


L J Rodr Íguez-Mu Ñiz

M L Ópez-D Íaz

On the exchange of iterated expectations of random upper semicontinuous functions

Keywords: Hukuhara differentiability, iterated expectation, Kudo-Aumann's integral, random upper semicontinuous function

Results about the exchange of iterated expectations of random upper semicontinuous functions, when measurability on product spaces is not necessarily assumed, are obtained.

Introduction

Reversing the order of iterated integrals has been an important topic in Probability and Measure Theories. The existence of a wide variety of situations to be discussed, depending on the considered mappings, especially on the space in which these mappings take on values, as well as on the considered concept of integral, has led to a great number of works on this topic. Most of these studies assume that the mapping to be integrated is measurable on the product space. However, there are situations in which this measurability is not satisfied, but iterated integrals are well-defined (see for instance [START_REF] Friedman | A consistent Fubini-Tonelli theorem for nonmeasurable functions[END_REF].

In this paper we obtain conditions which allows us to exchange iterated expectations of random upper semicontinuous functions without requiring the measurability of such random elements on the product space. The main mathematical complexities to obtain these results will be focused on the values these elements take on, the concept of integral for such mappings and the lack of measurability on product spaces.

A c c e p t e d m a n u s c r i p t

The structure of the paper is as follows: preliminary concepts are gathered in Section 2; the main results concerning the exchange of iterated expectations of random upper semicontinuous functions are given in Section 3; to conclude, Section 4 contains some illustrative examples.

Preliminaries

Let K c denote the class of nonempty compact convex subsets of R. This class can be endowed with a semilinear structure given by the Minkowski addition and the product by a scalar, that is,

M + N = {m + n | m ∈ M, n ∈ N } and λM = {λm | m ∈ M } with λ ∈ R, M, N ∈ K c .
The Hausdorff metric on K c is given by

d H (M, N ) = max { | inf M -inf N |, | sup M -sup N | }.
The space (K c , d H ) is complete and separable (see [START_REF] Debreu | Integration of correspondences[END_REF].

Given M ∈ K c , its magnitude, denoted by denoted by M , is defined as If S : Ω → K c is a random set, the mapping S : Ω → R, given by S (ω) = S(ω) for all ω ∈ Ω, is measurable (see [START_REF] Hiai | Integrals, conditional expectations and martingales of multivalued functions[END_REF].

d H (M, {0}) = sup x∈M |x|. If (Ω, A) is a measurable space, a mapping S : Ω → K c is said to be a random set if it is A|B d H -measurable,
If µ : A → R is a measure, a random set S is said to be integrably bounded with respect to µ, if S ∈ L 1 (Ω, A, µ).

The integral, or expected value in case of µ being a probability, of S, is given by the Kudo-Aumann integral (see for instance Hiai and Humegaki, 1977), that is,

Ω f (ω) dµ(ω) | f : Ω → R, f ∈ L 1 (Ω, A, µ), f ∈ S a.e. [µ] .
This set will be denoted by Ω S(ω) dµ(ω) or E(S|µ). It is easy to see that

E(S|µ) = Ω inf S(ω) dµ(ω), Ω sup S(ω) dµ(ω) .
Let F c denote the class of upper semicontinuous functions U : R →

[0, 1] such that U α ∈ K c for all α ∈ [0, 1], where U α = {x ∈ R : U (x) ≥ α} for

A c c e p t e d m a n u s c r i p t

α ∈ (0, 1], and U 0 = cl {x ∈ R : U (x) > 0}, cl denoting the topological closure. These mappings are also referred to as fuzzy sets of R, and the sets U α as the associated α-levels.

The class F c can be endowed with a semilinear structure, where addition and product by a scalar can be defined by means of Zadeh's extension principle (see [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning[END_REF], or equivalently (see [START_REF] Puri | Différentielle d'une fonction floue[END_REF] these operations can be levelwise calculated as

(U + V ) α = U α + V α and (λU ) α = λU α for all U, V ∈ F c , λ ∈ R and α ∈ [0, 1].
On F c we consider the d ∞ metric (see [START_REF] Puri | Différentielle d'une fonction floue[END_REF] given by

d ∞ (U, V ) = sup α∈[0,1] d H (U α , V α ), with U, V ∈ F c . The magnitude of U ∈ F c is given by U = d ∞ (U, 1 {0} ) = d H (U 0 , {0}).
Given a measurable space (Ω, A), a mapping X : Ω → F c is said to be a random upper semicontinuous function (r.u.s.f. for short) if X α : Ω → K c with X α (ω) = (X(ω)) α for all ω ∈ Ω, is a random set for all α ∈ [0, 1] (see Puri and[START_REF] Puri | Fuzzy random variables[END_REF][START_REF] Colubi | A D E [0, 1] representation of random upper semicontinuous functions[END_REF].

An r.u.s.f. X is said to be integrably bounded with respect to a measure

µ : A → R, if the mapping X ∈ L 1 (Ω, A, µ), where X : Ω → R is given by X (ω) = X(ω) for all ω ∈ Ω.
For an integrably bounded r.u.s.f., [START_REF] Puri | Fuzzy random variables[END_REF] define its integral, denoted by Ω X(ω) dµ(ω) or E(X|µ), as the unique set in F c such that E(X|µ) α = E X α µ for every α ∈ [0, 1]. When Ω = [a, b], we will use also the notation b a X(ω) dµ(ω). If µ is a probability measure, an r.u.s.f. is also referred to as a fuzzy random variable and its integral as the fuzzy expected value of X.

It is possible to extend to upper semicontinuous functions the concept of Hukuhara difference or Minkowski difference for subsets (see [START_REF] Hukuhara | Intégration des applications mesurables dont la valeur est un compact convexe[END_REF], so given U, V ∈ F c , its Hukuhara difference, denoted by

U -h V , is the set W ∈ F c (if it exists) such that U = V + W . Let T be a nonempty open subset of R. A mapping G : T → F c is said to be Hukuhara differentiable at t 0 ∈ T if there exists G (t 0 ) ∈ F c , which is called the Hukuhara differential of G at t 0 , such that lim ∆t→0 + d ∞ G(t 0 + ∆t) -h G(t 0 ) ∆t , G (t 0 ) = lim ∆t→0 + d ∞ G(t 0 ) -h G(t 0 -∆t) ∆t , G (t 0 ) = 0.

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The above definition (see [START_REF] Puri | Differentials of fuzzy functions[END_REF]) is an extension of the Hukuhara's differentiability criterion for set-valued mappings [START_REF] Hukuhara | Intégration des applications mesurables dont la valeur est un compact convexe[END_REF].

If a mapping G depends on more than one argument, we will make use of the usual symbol of partial derivative to indicate with respect to which argument the Hukuhara differential is considered.

Exchange of iterated expectations of random upper semicontinuous functions

In this section, the main results on the exchange of iterated expectations of r.u.s.f. with respect to probability distributions depending on certain families of parameters are developed. Throughout the paper, for any set

Ω ⊂ R k with k ∈ N, B Ω will denote the Borel σ-field on Ω. Given (Ω, B Ω ) and m 1 , m 2 : Ω → R two σ-finite measures, m 1
m 2 will indicate that m 1 is absolutely continuous with respect to m 2 , and dm 1 dm 2 will denote a Radon-Nikodym derivative of m 1 with respect to m 2 . If it is supposed that there exists a continuous Radon-Nikodym derivative, then dm 1 dm 2 will denote this particular function. Firstly, we prove a result on Radon-Nikodym derivatives, which extends a classical result for real-valued functions to the case of r.u.s.f.

Proposition 3.1. Let Ω ⊂ R k and let m 1 , m 2 : B Ω → R be two σ-finite measures with m 1 m 2 . If F : Ω → F c is an integrably bounded r.u.s.f. with respect to m 1 , then Ω F (s) dm 1 (s) = Ω F (s) dm 1 dm 2 (s) dm 2 (s) .
Proof. Obviously, F dm 1 dm 2 is an r.u.s.f. since dm 1 dm 2 is a measurable mapping (see [START_REF] Puri | Fuzzy random variables[END_REF]. On the other hand

F (s) dm 1 dm 2 (s) = F (s) dm 1 dm 2 (s), whence Ω F (s) dm 1 dm 2 (s) dm 2 (s) = Ω F (s) dm 1 (s). Since F ∈ L 1 (Ω, B Ω , m 1 ) then F dm 1 dm 2 ∈ L 1 (Ω, B Ω , m 2 ), and so F dm 1 dm 2
is integrably bounded with respect to m 2 .

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Consider now for each α ∈ [0, 1] the random set F α , and the random variables inf F α and sup F α . Then, we have that

Ω F α (s) dm 1 (s) = Ω inf F α (s) dm 1 (s), Ω sup F α (s) dm 1 (s) = Ω inf F α (s) dm 1 dm 2 (s) dm 2 (s), Ω sup F α (s) dm 1 dm 2 (s) dm 2 (s) = Ω F α (s) dm 1 dm 2 (s) dm 2 (s) = Ω (F (s) dm 1 dm 2 (s)) α dm 2 (s),
which concludes the proof.

The following result generalizes the First Fundamental Theorem of Calculus for the case of r.u.s.f. and the Hukuhara differential.

Proposition 3.2. Let [a, b], B [a,b] , m 1 be a σ-finite measure space with m 1
m, where m stands for the Borel measure on the interval

[a, b]. Let F : [a, b] → F c be an integrably bounded r.u.s.f. with respect to m 1 , such that F dm 1 dm is continuous. We define G : [a, b] → F c with G(t) = t a F (s) dm 1 (s), then i) G is Hukuhara differentiable with G (t) = F (t) dm 1 dm (t) for all t ∈ (a, b), ii) G is an r.u.s.f. Proof. By Proposition 3.1, G(t) = t a F (s) dm 1 dm (s) dm(s) for all t ∈ [a, b]. Since F dm 1
dm is continuous, in accordance with Theorem 10.3.12 in [START_REF] Diamond | Metric Spaces of Fuzzy Sets: Theory and Applications[END_REF], G is Hukuhara differentiable and G (t) = F (t) dm 1 dm (t) for all t ∈ (a, b). The statement ii) is obvious by the Hukuhara differentiability of G.

The following technical result will be applied in subsequent propositions.

f : Ω × [a, b] → R with Ω ⊂ R k , such that i) for every ω ∈ Ω, f ω : [a, b] → R, with f ω (t) = f (ω, t), is measurable, ii) for every t ∈ [a, b], the mapping f t : Ω → R, with f t (ω) = f (ω, t),
Let ∆ t : Ω → R with ∆ t (ω) = t a f (ω, s) dm(s) for all ω ∈ Ω. Then ∆ t ∈ L 1 (Ω, B Ω , P ) for every t ∈ [a, b].
Proof. The Dominated Convergence Theorem implies the continuity of ∆ t , and so its measurability. The integrability is trivially obtained from iii).

Regarding the Hukuhara differentiability of r.u.s.f. under the integral sign, we can state the following results. i) for every ω ∈ Ω, the mapping

X ω : [a, b] → F c , with X ω (t) = X(ω,
∂ ∂t Ω t a X(ω, s) dP ω (s) dP (ω) = Ω ∂ ∂t t a X(ω, s) dP ω (s) dP (ω) and hence ∂ ∂t Ω t a X(ω, s) dP ω (s) dP (ω) = Ω X(ω, t) dP ω dm (t) dP (ω),
which concludes the proof.

The following result proves the continuity under the integral sign for a special class of mappings. Proposition 3.5. Let (Ω, B Ω , P ) be a probability space with Ω ⊂ R k , let T be a convex subset of R and t 0 ∈ T . For every t ∈ T , let P t be a probability measure on (Ω, B Ω ) such that P t P . If X : Ω × T → F c satisfies that i) for every t ∈ T , the projection X t is an integrably bounded r.u.s.f. with respect to P t , ii) the mapping t → X(ω, t) dPt dP (ω) is continuous at t 0 , a.s.

[P ],

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iii) there exists h ∈ L 1 (Ω, B Ω , P ) and N a neighborhood of t 0 , such that for

all t ∈ N ∩ T X(ω, t) dP t dP (ω) ≤ h(ω) a.s. [P ],
then, ∆ :

T → F c , with ∆(t) = Ω X(ω, t) dP t (ω), is continuous at t 0 . Proof. Let {t n } n∈N ⊂ T with lim n t n = t 0 , Proposition 3.1 implies that ∆(t n ) = Ω X(ω, t n ) dPt n dP (ω) dP (ω)
for all n ∈ N, so the Dominated Convergence Theorem for r.u.s.f. (see [START_REF] Puri | Fuzzy random variables[END_REF] proves the result.

The following theorem contains the key result in the paper.

Theorem 3.6. Let (Ω, B Ω , P ) be a probability space with Ω ⊂ R k and let m denote the Borel measure on the interval T = [a, b]. For every t ∈ T , let P t be a probability measure on (Ω, B Ω ) such that P t P and there exists a continuous Radon-Nikodym derivative. For every ω ∈ Ω, let P ω be a probability on (T, B T ) such that P ω m and there exists a continuous Radon-Nikodym derivative.

Let X : Ω × T → F c be a mapping satisfying that: i) for every t ∈ T , X t is an integrably bounded r.u.s.f. with respect to P t , ii) for every ω ∈ Ω, X ω is an integrably bounded r.u.s.f. with respect to P ω and it is continuous a.s.

[P ],

iii) there exists h 1 ∈ L 1 (Ω, B Ω , P ) such that X(ω, t) dPω dm (t) ≤ h 1 (ω) a.s. [P ] for every t ∈ T , and the mapping ω → X(ω, t) dPω dm (t) is continuous a.e. [m], iv) there exists a mapping g

∈ L 1 ([a, b], B [a,b] , m) such that for every ω ∈ Ω, X(ω, t) dPω dm (t) ≤ g(t) a.e. [m] for every ω ∈ Ω, v) the mapping t → X(ω, t) dPt dP (ω) is continuous on T a.s. [P ], vi) there exists h 2 ∈ L 1 (Ω, B Ω , P ) such that X(ω, t) dPt dP (ω) ≤ h 2 (ω) a.s. [P ] for every t ∈ T .
Let m be a probability measure on (T, B T ) such that m m and there exists a continuous Radon-Nikodym derivative. If for every t ∈ T , the equality

dPω dm (t) = dPt dP (ω) dm dm (t) a.s. [P ] holds, then Ω t a X(ω, s) dP ω (s) dP (ω) = t a Ω X(ω, s) dP s (ω) dm (s)

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for every t ∈ T .

Proof. The case t = a is trivial. Consider now t ∈ (a, b). On the basis of Proposition 3.2 we deduce that the mapping t → t a X(ω, s) dP ω (s) is Hukuhara differentiable a.s. [P ], with differential X(ω, t) dPω dm (t). In virtue of Proposition 3.4 the mapping t → Ω ( t a X(ω, s) dP ω (s))dP (ω) is Hukuhara differentiable with differential given by

∂ ∂t Ω t a X(ω, s) dP ω (s) dP (ω) = Ω X(ω, t) dP ω dm (t) dP (ω).
On the other hand, by means of Proposition 3.5, the mapping on [a, b] given by s → Ω X(ω, s)dP s (ω) is continuous, and applying Propositions 3.1 and 3.2 to this mapping, we obtain that Therefore, we have obtained that the Hukuhara differentials of the above mappings coincide for all t ∈ [a, b), that is,

∂ ∂t t a Ω X(ω, s)dP s (ω) dm (s) = Ω X(ω,
∂ ∂t Ω t a X(ω, s) dP ω (s) dP (ω) = ∂ ∂t t a Ω X(ω, s) dP s (ω) dm (s)
for every t ∈ [a, b). Since these mappings have the same Hukuhara differentials, they should be equal except for a constant, but when t = a the mappings are equal, so they must coincide for t ∈ [a, b).

When t = b, it is sufficient to consider a sequence {t n } n∈N with lim n t n = b, and applying the Dominated Convergence Theorem for r.u.s.f. Finally, the above theorem can be extended to unbounded intervals as follows,

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Theorem 3.7. Assume the conditions in Theorem 3.6 with the interval T being not necessarily bounded, and suppose that there exists g ∈ L 1 (Ω, B Ω , P ) such that T X(ω, s) dP ω (s) ≤ g (ω) a.s. [P ]. Then, the following equality holds,

Ω T X(ω, s) dP ω (s) dP (ω) =
T Ω X(ω, s) dP s (ω) dm (s) .

Proof. Consider two sequences {a n } n∈N , {b n } n∈N ⊂ T such that a n ≤ b n for all n ∈ N, lim n a n = inf T and lim n b n = sup T, where inf T and sup T can be either finite or infinite. On the basis of Theorem 3.6, we deduce that

Ω bn an X(ω, s) dP ω (s) dP (ω) = bn an Ω X(ω, s) dP s (ω) dm (s),
and so, the Dominated Convergence Theorem proves the result.

Remark 3.8. We should note that the measurability and continuity conditions assumed in the above results do not imply that X is an r.u.s.f. with respect to the product measurable space (Ω × T, B Ω ⊗ B T ). Thus, consider Ω = T = [0, 1] and C the Cantor set. It is well known that C ∈ B [0,1] and there exists N ∈ B [0,1] with C ⊂ N and m(N ) = 0. Consider the mapping f : R → R 2 with f (x) = (x, x) and

A = f (C). Obviously, f is B [0,1] |B [0,1] ⊗ B [0,1] measurable since it is continuous. If we define g : [0, 1] × [0, 1] → R with g(ω, t) = 1 A (ω,
[0,1] ⊗ B [0,1] |B [0,1] measurable, then g • f | [0,1] would be measurable, but (g • f ) -1 ({1}) = C ∈ B [0,1] .
Thus, if we define X : [0, 1] × [0, 1] → F c with X(ω, t) = 1 {g(ω,t)} , then X satisfies all measurability and continuity conditions in the above theorems, but trivially it is not an r.u.s.f. on the product measurable space.

Illustrative examples

In this section we propose some illustrative examples in relation with the theoretical results developed in the above Section. ) absolutely continuous with respect to the Borel measure on [0, 1], with continuous density functions.

Let P t = P 1 for all t ∈ [0, 1], P = P 1 , P ω = P 2 for all ω ∈ [0, 1], and m = P 2 . It is obvious that P t P, the constant equals 1 being a continuous Radon-Nikodym derivative. By hypothesis P ω = P 2 m and there exists a continuous density (Radon-Nikodym derivative).

Let X : [0, 1] × [0, 1] → F c with X bounded, satisfying that X ω is an r.u.s.f., and continuous a.s. [P 1 ] while X t is an r.u.s.f., and continuous a.s.

[m] (note that the mapping in Remark 3.8 satisfies such conditions), then Theorem 3.6 implies that It is interesting to notice that the expression [0,1]×[0,1] X(ω, s) dP 1 (ω) ⊗ P 2 (s), P 1 ⊗P 2 being the product probability, has no sense since X is not measurable on the product space.

Concluding remarks

To conclude we should indicate that results which guarantee the exchange of iterated expectations of random upper semicontinuous functions when these are measurable on product spaces can be found in [START_REF] López-Díaz | Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications[END_REF], where on the basis of such results, the authors develop some statistical applications. The results in this paper suggest the idea of analyzing analogous statistical applications for random upper semicontinuous functions when they are not measurable on the product space. Especial importance could have these results in the Bayesian analysis of the single stage decision problem when utilities are nonmeasurable function-valued mappings, since an exchange of integrals similar to the one given in Theorem 3.6 and 3.7 is necessary.

  where B d H denotes the Borel σ-field generated by the topology induced by d H on K c . It is known that S is a random set if and only if the mappings inf S, sup S : Ω → R are measurable when we consider on R the usual Borel σ-field.

Lemma 3. 3 .

 3 Let (Ω, B Ω , P ) be a probability space with Ω ⊂ R k and let m stand for the Borel measure on the interval [a, b]. Consider a mapping

  exists g ∈ L 1 ([a, b], B [a,b] , m) such that |f (ω, t)| ≤ g(t) a.e.[m] for every ω ∈ Ω.

Proposition 3. 4 .

 4 Let (Ω, A, P ) be a probability space with Ω ⊂ R k , and let m denote the Borel measure on the interval [a, b]. For every ω ∈ Ω, let P ω be a probability measure on ([a, b], B [a,b] ) with P ω m, such that there exists a continuous Radon-Nikodym derivative. If X : Ω × [a, b] → F c satisfies the following conditions:

  Example 4.2. Let us consider [a, b] = Ω = [0, 1]. Let P 1 and P 2 be probabilities on ([0, 1], B [0,1]

  , s) dP 1 (ω) dP 2 (s) = , s) dP 2 (s) dP 1 (ω).

  Let us see that the projection H t is an integrably bounded r.u.s.f. with respect to P for every t ∈ [a, b]. Let t ∈ [a, b] and let us consider the α-level sets of the mapping H t (ω), then H t (ω) α Muñiz and López-Díaz (2003) applied to H, and the condition ii) in the statement of this proposition, imply that

	It is easy to check that the mappings on Ω × [a, b] given by
	(ω, s) → inf X(ω, s)	dP ω dm	(s)	α	and			(ω, s) → sup X(ω, s)	dP ω dm	(s)	α
	fulfill conditions in Lemma 3.3 for every α ∈ [0, 1]. Therefore, the mappings
	H t (ω) α are integrably bounded random sets for every α ∈ [0, 1], and hence H t (ω) is an integrably bounded r.u.s.f. with respect to [P ] for all t ∈ [a, b].
	On the other hand, on the basis of Proposition 3.2, the continuity of
	∂ ∂t Ω Proof. The hypothesis i) implies that H(ω, t) = t a X(ω, s) dP ω (s) dP (ω) = Ω fined. By Proposition 3.1 we obtain that H(ω, t) = X(ω, t) t a X(ω, s) dP ω (s) is well de-dP ω dm (t) dP (ω). t a X(ω, s) dPω dm (s) dm(s). A c c e p t e d m dPω dm , and that of X ω a.s. [P ], we deduce that the projection H ω is Hukuhara differentiable for every t ∈ (a, b) a.s. [P ], with ∂ ∂t H(ω, t) = X(ω, t) dP ω dm (t) a.s. [P ] . a n u s c r i p t Theorem 4.4 in Rodríguez-
	=	a	t	inf(X(ω, s)	dP ω dm	(s)) α dm(s),	a	t	sup(X(ω, s)	dP ω dm	(s)) α dm(s) .

t), is an integrably bounded r.u.s.f. with respect to P ω , and X ω is continuous a.s. [P ],

ii) there exists h ∈ L 1 (Ω, A, P ), such that X(ω, t) dPω dm (t) ≤ h(ω) a.s. [P ] for every t ∈ [a, b], and the mapping ω → X(ω, t) dPω dm (t) is continuous a.e. [m], iii) there exists g ∈ L 1 ([a, b], B [a,b] , m) with X(ω, t) dPω dm (t) ≤ g(t) a.e. [m] for every ω ∈ Ω, then, the mapping t ∈ [a, b] → Ω t a X(ω, s) dP ω (s) dP (ω) is Hukuhara differentiable on (a, b), and for every t ∈ (a, b) it holds that

  t) (where 1 stands for the indicator function) then g t and g ω are measurable for all t, ω ∈ [0, 1]. Moreover, both are continuous for almost every t [m] and almost every ω [m] respectively. However, if g were B

For each ω ∈ (0, 1) take the probability P ω on ([0, 1], B [0,1] ) whose density function is given by h ω (t) = (-2t/ω + 2ω -1/2 )1 (0,ω 1/2 ) (t).

Given t ∈ [0, 1] we consider the probability P t on ((0, 1), B (0,1) ) with density function equal to h t (ω) = (2/(1-t 1/2 ) 2 ω-2t -1/2 /(1-t 1/2 ) 2 )1 (t 1/2 ,1) (ω) if t = 1, whereas if t = 1 we consider the uniform distribution on (0, 1).

Let X : (0, 1) × [0, 1] → F c given by the mapping in Remark 3.8. Let us see that all conditions in Propositions 3.4 and 3.5 are satisfied. In relation to the former, it is obvious that P ω m and there exists a continuous Radon-Nikodym derivative (note that h ω is continuous in [0, 1]) for all ω ∈ (0, 1).

On the other hand,

so X is an r.u.s.f. Since X ≤ 1, we obtain that X is integrably bounded with respect to any probability, and clearly X ω is continuous a.s. [P ].

Moreover, |h ω (t)| ≤ 2/ω 1/2 ∈ L 1 (0, 1), B (0,1) , P ), and the mapping ω → X(ω, t)h ω (t) is continuous a.s. [m].

With respect to the condition iii) in Proposition 3.4, it holds that

As a consequence, we conclude that the mapping

is Hukuhara differentiable on (a, b), and

In relation to Proposition 3.5, considering T = [0, 1], then P t m for all t ∈ [0, 1]. Observe that there exists a continuous Radon-Nikodym derivative given by h t .

It is trivial that the condition i) of Proposition 3.5 is held. On the other hand, the mapping t → X(ω, t)h t (ω) is continuous in any point t 0 ∈ [0, 1] a.s. [P ]. Moreover X(ω, t)h t (ω) ≤ 2ω for all ω ∈ (0, 1), this mapping belonging to L 1 (0, 1), B (0,1) , m .

Hence, the mapping : T → F c with (t) =


René Blacher

Central Limit Theorem by Moments

Keywords: Central Limit Theorem, moments, strongly mixing sequence, higher order correlation coefficients

In a previous central limit theorem by moments, it has been proved that the moments converge to those of the normal distribution if the moments of sums are asymptotically independent (cf Theoreme de la limite centrale par les moments, Blacher R. Cpte rendus Acad. Sci Paris. t-311-I, 465-468). . In this paper we generalize this result by adding a negligible sequence to these sums. So, we can prove that the moments of some functionals of strong mixing sequences converge.

Introduction

Let {X n } be a sequence of random variables defined on a probability space (Ω, A, P ) such that, for all n ∈ N , E{X n } = 0 and 0 < E{|X n | k } < ∞ for all k ∈ N .

On decompose X 1 +X 2 +.........+X n en X 1 +X 2 +.........+X u(n) , X u(n)+1 + X u(n)+2 + ............ + X u(n)+t(n) and X u(n)+t(n)+1 + X u(n)+t(n)+2 + ......... + X u(n)+t(n)+u (n) , where u(n) + t(n) + u(n) = n. Notations 1.1 : We denote by κ(n) ∈ N , an increasing sequence such that κ(1) = 0, κ(n) ≤ n and κ(n)/n → 0 as n → ∞ . We define the sequences u(n) and t(n) by : u(1)=1, u(n) = max m ∈ N * 2m + κ(m) ≤ n and t(1)=0, t(n) = n-2u(n) if n ≥ 2. Moreover, we simplify u(n) and t(n) as u and t.

Let σ(u) 2 be the variance of X 1 +X 2 +...+X u . One sets S u = X1+X2+.........+Xu σ (u) , ξ u = Xu+1+Xu+2+.........+Xu+t σ (u) and S u = Xu+t+1+Xu+t+2+.........+Xu+t+u σ (u) . If t(n)=0, we set ξ u = 0. Moreover, one tolerates the notation S n = X1+X2+.........+Xn σ (n) .

We denote also by σ o (t) 2 the variance of X u+1 + X u+2 + ......... + X u+t .

A c c e p t e d m a n u s c r i p t

In [?] , one has proved that u → ∞, n/u → 2 and t/u → 0 as n → ∞.

Moreover, if κ(n) → ∞, t → ∞ as n → ∞. Finally, ξ u is negligible if σ o (t)/σ(u) → 0 : E{(ξ u ) 2 } → 0 .
Notations 1.2 : We define conditions H mS and H mI by the following way.

H mS : ∀p ∈ N , E (S u + v u ) p -E (S u + v u ) p → 0 as n → ∞ . H mI : ∀(p, q) ∈ N 2 , E (S u + v u ) p (S u + v u ) q -E (S u + v u ) p E (S u + v u ) q → 0 as n → ∞ ,
where {v u } and {v u } are two sequences of random variables such that . But they did not studied the moments's convergence. Now another look is possible : one can define asymptotical independence conditions by using moments. In this way, one can use higher order correlation coefficients (cf Lancaster [?], Blacher [?]). Then, in [?] we have turned the convergence of moments into an equivalent condition on these coefficients. These results have led to the introduction of the condition H mI : that is, rather than to assume that {X n } is strong mixing, we choose H mI as asymptotical independence assumptions. Then, under this assumption, S n → M N (0, 1) , that is, for all p ∈ N , E{(S n ) p } → µ p as n → ∞ , the moment of order p of N(0,1).

E{(v u ) 2k }+ E{(v u ) 2k } → 0 as n → ∞ for all k ∈ N . Now,
Théorème 1 : Assume that E{|ξ u | k } → 0 as n → ∞ for all k ∈ N . Assume that H mS and H mI hold. Then, S n → M N (0, 1).
In [?], we have proved this theorem when v u = v u = 0. In section III, we prove theorem ?? for any sequences v u and v u . In II-a, we explain why we need this generalization. In II-b, we apply these results to functionals of IID sequences. In II-C we prove the MCLT for some functionals of strong mixing sequences or more general sequences.

A c c e p t e d m a n u s c r i p t

2 Moments Central Limit Theorems for functionals of some strictly stationary sequences

Discussion

Notations 2.1 : Let Ψ t be a strictly stationary process. Assume that

X t = ∞ i=0 k i+1 (Ψ t+i )
, where the k i 's are measurable functions such that E{k i (Ψ 1 )} = 0 and 0 < E{k i (Ψ 1 ) 2p } < ∞ for all i ∈ N and all p ∈ N .

We set M n = 1 σ(n) n r=1 r s=1 k s (Ψ r ) and r n = 1 σ(n) ∞ r=1 r+n s=r+1 k s (Ψ n+r ). Then, one can write S n = M n + r n with E{M n } = E{r n } = 0 . Of course, S u(n) = M u(n) + r u(n) . Moreover, one can write S u(n) = M u(n) + r u(n) with M u = 1 σ(u) u r=1 r s=1 k s (Ψ r+u+t ) and r u = 1 σ(u) ∞ r=1 r+u s=r+1 k s (Ψ u+r+u+t ) .
Now one wants to know if the MCLT hold. For example let {Θ t } be an IID sequence of random variables. Assume that Ψ t = Θ t and that, for all p ∈ N ,

E{|r n | p } → 0 as n → ∞ .
In order to prove the MCLT, one can try to apply the results of [START_REF] Blacher R | Theoreme de la limite centrale par les moments[END_REF], i.e. theorem ?? with v u = v u = 0 : we write

E{(S u ) p (S u ) q } = E{(M u ) p }E{(M u ) q }+ s =q,r =p A r,s E (M u ) r (r u ) p-r (M u ) s (r u ) q-s . By Holder Inequality, E (M u ) r (r u ) p-r (M u ) s (r u ) q-s converges to 0 if E{(M u ) r } is bounded.
Then, in order to use the theorem ?? with v u = v u = 0, we have to prove that E{(M u ) r } is bounded. It is also difficult as well to prove that E{(S n ) r } is bounded. But, if one uses theorem ?? with v u = -r u and v u = -r u , H mI holds obviously : E (S u +v u ) p (S u +v u ) q = E (S u +v u ) p E (S u +v u ) q . For this reason we generalize theorem ?? to the case where v u and v u are not equal to 0.

II-b) Functional of some IID sequences.

Proposition 2.1 : Assume that Ψ t = Θ t and that

∞ s=0 ∞ i=s k i+1 (Ψ 1 ) < C < ∞ . Then, S n → M N (0, 1).
Proof : First, we need the following lemma.

Lemma 2.2 : Assume that E k

i (Ψ r )k j (Ψ s ) = 0 for all (i, j, r, s) ∈ N 4 , r = s. Assume ∞ s=0 ∞ i=s k i+1 (Ψ 1 ) < C < ∞ and that, in L 2 (Ω) , r i=1 k i (Ψ 1 ) → K as r → ∞ where K 2 .dP > 0. Then, σ(n) → ∞ and sup|r n | → 0 . Proof : There exists N 0 such that E n r=1 r s=1 k s (Ψ r ) 2 = n r=1 E r s=1 k s (Ψ r ) 2 ≥ (n/2) K 2 .dP if n ≥ N 0 . Moreover, σ(n)|r n | ≤ 2C.

A c c e p t e d m a n u s c r i p t

We choose κ(n) = 0. Because σ(n) → ∞ , E{|ξ u | p } → 0 as n → ∞ for all p ∈ N . Moreover, E{(M u ) p } = E{(M u ) p } and E{(M u ) p (M u ) q } = E{(M u ) p }E{(M u ) q }. Then we apply theorem ?? with v u = -r u and v u = -r u .

Example 2.3 : Assume that Θ 1 has the standard normal distribution and that k i (Ψ 1 ) = i -3 sin(2πΘ 1 ). Then, S n → M N (0, 1).

One can also prove the asymptotic normality in some cases where r n is not bounded .

Example 2.4 :

AR(1)Process. Assume X t = ∞ i=0 a i Θ t+i where 0 < a < 1 . Then, M n = 1 σ(n) n j=1 1-a j 1-a Θ j and r n = 1 σ(n) ∞ j=n+1 a j-n (1-a n ) 1-a Θ j .
Therefore, for all r ∈ N , κu r (r u ) = κur(Θ1)

σ(u) r ∞ j=u+1 a j-u (1-a u ) 1-a r → 0 ,
where κu r is the r-th cumulant. Therefore, E{|r u | p } → 0 for all p ∈ N * . Then, the condition H mI is satisfied for any sequence {u(n)} defined in notation ??. Then, S n → M N (0, 1) .

Recall, there exists AR(1) processes which are not strongly mixing : e.g. a=1/2, P {Θ t = 1/2} = P {Θ t = -1/2} = 1/2(cf [?] p 360-362). Then, these processes satisfy H mI but are not strongly mixing. Moreover, S n → M N (0, 1).

Functional of some strong mixing sequences

Strong mixing sequences can often be approached by functional of IID sequences. For example, let

τ t = ∞ i=0 b i+1 h i (Θ t+i )
, where |h i (Θ 1 )| ≤ 1 , |b| ≤ 1/2 and where Θ t is an IID sequence independent of Θ t .

For example, if h i (Θ 1 ) = a i Θ 1 , τ t can be strong mixing or not. Then, one can prove MCLT even for functionals of some non strong mixing sequence.

Example 2.5 : Assume that k

i (Ψ t ) = i -5/2 g i (Θ t ).sin(e(i)τ t ) where E{g i (Θ 1 )} = 0 and |e(i)| ≤ 2π for all i ∈ N . Assume that ∞ s=0 ∞ i=s k i+1 (ψ 1 )| < C < ∞. Then, by lemma ??, σ(n) → ∞ and sup|r n | → 0 as n → ∞ . Moreover, |τ t | ≤ 1 and τ t = τ t,u-t +b u-t+1 ρ where |ρ| ≤ 1 and τ t,u-t = u-t i=0 b i+1 h i (Θ t+i ). Therefore, sin(e(s)τ r ) = sin(e(s)τ r,u-r ) + 2πb u-r+1 ρ * where |ρ * | ≤ 1.
Then, M u = σ(u) -1 u r=1 r s=1 s -5/2 g s (Θ r )sin(e(s)τ r,u-r ) + r * u , where Sup|r * u | → 0 as n → ∞. Then, one applies theorem ?? with v u = -r ur * u and a similar decomposition for M u :

E (S u + v u ) p (S u + v u ) q = E (S u + v u ) p E (S u + v u ) q .
One deduces that S n → N (0, 1) .

By using theorem ??, the MCLT can be proved for many other functionals, in particular , if {Ψ t } is a q-dependent sequence.

E{S u ξ u } 2 ≤ E{S 2 u }E{ξ 2 u } ≤ E{ξ 2 u } = o(1)
where "o" is the classical "o" : o(1) → 0 as n → ∞ . We deduce that σ(n

) 2 = σ(u) 2 E{(S u + v u + S u + v u + ξ u -v u -v u ) 2 } = σ(u) 2 [2E{(S u ) 2 } + o(1)]. Then, for all h ∈ N , σ(u) h /σ(n) h = 2 -h/2 + o(1).
For all p ∈ N , we set

M p n = E{(S n ) p }.Clearly, M 0 n = 1 ,M 1 n = 0 and M 2 n = 1. Let h ∈ N , 2 ≤ h ≤ 2k. Assume that, for all p ∈ N , p ≤ 2h -2, M p
n → µ p , the moment or order p of N(0,1) .

By Holder inequality,

|E{(S u ) r (v u ) p-r }| ≤ E{|S u | p } r/p E{|v u | p } (p-r)/p if r < p. Therefore, if r < p ≤ 2h -2, E{(S u ) r (v u ) p-r } = o(1) . If r < 2h, E{(S u ) r (v u ) 2h-r } = M 2h s o(1), where M 2h s (n) = sup M 2h u(n) , 1 .
by using H mI and H mS ,

E (S u + v u + S u + v u ) 2h = E{(S u +v u ) 2h }+E{(S u +v u ) 2h }+ 2h-1 m=1 2h! m!(2h -m)! E{(S u +v u ) 2h-m }E{(S u +v u ) m }+o(1) = 2E{(S u + v u ) 2h } + 2h-2 m=2 2h! m!(2h -m)! E{(S u + v u ) 2h-m }E{(S u + v u ) m } + o(1) = 2E{S 2h u }+2 2h-1 r=0 2h! r!(2h -r)! E{S r u }E{v 2h-r u }+ 2h-2 m=2 2h! m!(2h -m)! E{S 2h-m u }E{S m u }+o(1) = 2M 2h u + M 2h s o(1) + 2h-2 m=2 2h! m!(2h -m)! µ 2h-m µ m + o(1) ≤ 2M 2h u + M 2h s |o(1)| + 2(2 h-1 -1)µ 2h + |o(1)|. Moreover, if m ≥ 1, |E{(S u + v u + S u + v u ) 2h-m (ξ u -v u -v u ) m }| ≤ |E{(S u + v u + S u + v u ) 2h }| (2h-m)/(2h) |E{(ξ u -v u -v u ) 2h }| m/(

A c c e p t e d m a n u s c r i p t 3

 3 Proof of theorem ??At first, for example, by Schwarz inequality,

  2h) ≤ |o(1)| 2M 2h s + M 2h s |o(1)| + 2(2 h-1 -1)µ 2h + |o(1)| (2h-m)/(2h) ≤ |o(1)|M 2h s + |o(1)|.

A c c e p t e d m a n u s c r i p t

  2h σ(n) 2h E{(S u + v u + S u + v u + ξ uv uv u ) 2h } = (2 -h + o(1)) 2h m=0 2h! m!(2hm)! E{(S u + v u + S u + v u ) 2h-m (ξ uv uv u ) m } ≤ 2 1-h M 2h s + M 2h s |o(1)| + (2 h-1 -1)µ 2h + |o(1)| + |o(1)|M 2h s + |o(1)| ≤ 2 1-h M 2h s + (2 h-1 -1)µ 2h [1 + (n)], where (n) → 0 as n → ∞.The same equality has been obtained in the proof of [?] when v u = v u = 0. We deduce by the same way that M 2h n is bounded. Therefore,M 2h n = 2 1-h M 2h u + (2 h-1 -1)µ 2h + o(1). We deduce the convergence of M 2h-1 n and M 2h n to µ 2h-1 and µ 2h .

  we want to study the MCLT (Moment Central Limit theorem) for {X n }. Generally, one assumes that {X n } is a martingale or is strong mixing. Then, one proves that under some assumptions, moments converge or are So one can deduce that H mI holds. For example, if{X n } is ϕ-mixing, H mI holds if t(n) → ∞ as n → ∞ enough slowly (cf [?],[?]). If {X n } is strong mixing, H mI holds if E{|S n | p } is bounded for all p ∈ N . Of course, if {X n } is strictly stationary, H mS holds.

	bounded : cf Bernstein, [?] , Brown [?] , Eissein-Janson [?], Hernndorf [?], Birkel
	[?], Krugov [?], Mairoboda [?], Yokohama ( [?],[?]), Cox-Kim [?], Ibragimov-
	Lifshits [?], Soulier [?], Rozovsky [?].
	Now, the martingale condition is a strong condition. It is the same for strong
	mixing conditions : even they does not hold for some AR(1) processes. Then,
	some authors have introduce weaker hypotheses : Versik Ornstein ([?], [?]),
	Withers [?], Cogburn [?] Rosenblatt [?], Pinskers [?]


Manuel Molina
email: mmolina@unex.es

Inés M Del Puerto
email: idelpuerto@unex.es

Alfonso Ramos
email: aramos@unex.es.

A class of controlled bisexual branching processes with mating depending on the number of progenitor couples

Keywords: branching processes, bisexual processes, controlled processes

We introduce a class of bisexual branching processes where a control is achieved on the number of couples and the mating depends on the number of progenitor couples. We establish several probabilistic properties and some results about its extinction probability.

Introduction

In order to describe the probabilistic evolution of two-sex populations, several classes of bisexual processes have been investigated, see e.g. [START_REF] Alsmeyer | The bisexual Galton-Watson process with promiscuous mating: extinction probabilities in the supercritical case[END_REF], [START_REF] Bruss | A note on extinction criteria for bisexual Galton-Watson processes[END_REF], [START_REF] Daley | Extinction conditions for certain bisexual Galton-Watson branching processes[END_REF], [START_REF] González | Limit behaviour for a subcritical bisexual Galton-Watson branching process with immigration[END_REF][START_REF] González | On the limit behaviour of a supercritical bisexual Galton-Watson branching process with immigration of mating units[END_REF], [START_REF] Molina | Bisexual Galton-Watson branching process with population-size dependent mating[END_REF] or [START_REF] Xing | On the extinction of one class of population-sizedependent bisexual branching processes[END_REF]. We refer the reader to [START_REF] Hull | A survey of the literature associated with the bisexual Galton-Watson branching process[END_REF] or [START_REF] Haccou | Branching processes: Variation, growth, and extinction of populations[END_REF] A c c e p t e d m a n u s c r i p t 2 M. Molina et al. for surveys about these processes. However, the range of bisexual models investigated is not large enough in order to get an optimum modelling in some populations. By environmental, social, or other factors, it could be advisable to carry out a control on the number of couples (female-male mating units) in the population. Furthermore it is reasonable to admit that, in different generations, the same number of females and males gives rise to different number of couples. In an attempt to contribute some solution to these situations, in this work we introduce the class of controlled bisexual branching processes with mating depending on the number of progenitor couples and we provide some theoretical results about it. In Section 2, we give its mathematical formal description and its intuitive interpretation. Section 3 is devoted to determining several probabilistic properties about the model and Section 4 deals with its extinction probability.

The model

We introduce the controlled bisexual process with mating depending on the number of progenitor couples as a two-sex model

{(F n , M n )} n≥1 initiated with Z * 0 = N * ≥ 1
couples and defined, for n ∈ Z + , in the recursive form:

(F n+1 , M n+1 ) = Z * n i=1 (f n,i , m n,i ), Z n+1 = L Z * n (F n+1 , M n+1 ), Z * n+1 = φ(Z n+1 ) (1)
where {(f n,i , m n,i )} n≥0;i≥1 is a sequence of independent and identically distributed non-negative, integer-valued random vectors; {L k } k≥0 is a sequence of non-negative real functions on R + × R + , for each k fixed L k is assumed to be monotonic nondecreasing in each argument, integer-valued on the integers, L k (x, 0) = L k (0, y) = 0, x, y ∈ R + ; and φ is a non-negative real function on R + such that φ(0) = 0 and Remark 2.1 The class of processes given in (1) includes some models investigated in the bisexual branching process literature. Indeed, when φ(x) = x, x ≥ 0 one obtains the class of bisexual models introduced in [START_REF] Molina | Bisexual Galton-Watson branching process with population-size dependent mating[END_REF]. We remark that if φ is non-decreasing then one has, for each

φ(k) ∈ Z + 0 , k ∈ Z + 0 ,
k ∈ Z + , that L * k = φ • L k is a mating function.
In such a particular case, the model ( 1) may be considered under the perspective of the class of processes studied in [START_REF] Molina | Bisexual Galton-Watson branching process with population-size dependent mating[END_REF] 1) is reduced to the class of models given in [START_REF] Daley | Extinction conditions for certain bisexual Galton-Watson branching processes[END_REF].

k (x, y) = L(x, y), x, y ∈ R + , k ∈ Z + then the class (
We will consider the following working assumptions about the mating functions:

(A1): For each k ∈ Z + , L k is a superadditive function, namely, for n ≥ 2 L k n i=1 x i , n i=1 y i ≥ n i=1 L k (x i , y i ), x i , y i ∈ R + , i = 1, . . . , n. (A2): For each x, y ∈ R + fixed, the sequence {L k (x, y)} k≥0 is non-decreasing.
Assumption (A1) expresses the fact that n i=1 x i females and n i=1 y i males coexisting together will form a number of couples greater than or equal to the total number of couples produced from n groups of x i females and y i males, i = 1, . . . , n, living separately. Assumption (A2) represents the usual fact of many biological populations in which the number of matings in certain generation depends on the number of progenitor couples in the previous one in such a way that the mating is promoted if such a number grows. Some sequences {L k } k≥0 verifying (A1) and (A2) are, for instance:

(a) L k (x, y) = x min{k, y}; (b) L k (x, y) = min{x, ky}; or (c) L k (x, y) = min{x, y} if k ≤ k 0 or x min{k, y} if k > k 0 (restricted to non-negative integers)
where k 0 ∈ Z + 0 is fixed.

Probabilistic properties

Let us denote by f (s,

t) = E[s f 0,1 t m 0,1 ], h n (s, t) = E[s F n t M n ] and g * n (s) = E[s Z * n ], s, t ∈ [0, 1], n ∈ Z + 0 . Clearly g * 0 (s) = s N * . Proposition 3.1 For s, t ∈ [0, 1] and n ∈ Z + , h n+1 (s, t) = g * n (f (s, t)) .

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Proof.

It is clear that h 1 (s, t) = g * 0 (f (s, t)), s, t ∈ [0, 1]. For n ∈ Z + 0 h n+1 (s, t) = E E[s Fn+1 t Mn+1 | Z n ] = ∞ k=0 E s P φ(k) i=1 fn,i t P φ(k) i=1 mn,i P (Z n = k) = ∞ k=0 (f (s, t)) φ(k) P (Z n = k)g * n (f (s, t)) , s, t ∈ [0, 1]. Let us write µ = E[(f 0,1 , m 0,1 )], Σ = Cov[(f 0,1 , m 0,1 )], µ n = E[(F n , M n )] and Σ n = Cov[(F n , M n )], n ∈ Z + 0 . As a consequence of Proposition 3.1, µ n+1 = E[Z * n ]µ and Σ n+1 = E[Z * n ]Σ + V ar[Z * n ]µ µ, n ∈ Z +
, where µ denotes the transpose of µ.

Let Y k = L k (f 0,1 , m 0,1 ) and Y * k = φ(Y k ), k ∈ Z + and let us denote by π k (s) = E[s Y k ], π * k (s) = E[s Y * k ], and g n+1 (s) = E[s Z n+1 ], s ∈ [0, 1], k, n ∈ Z + . Clearly Y 0 = Y * 0 = 0 hence π 0 (s) = π * 0 (s) = 1. Proposition 3.2 Assume (A1). For s ∈ [0, 1] and n ∈ Z + (i) g n+1 (s) ≤ E π Z * n (s) Z * n . (ii) If φ is non-increasing on (0, ∞) then g * n+1 (s) ≥ E (π * Z * n (s)) Z * n . (iii) If φ is superadditive then g * n+1 (s) ≤ E (π * Z * n (s)) Z * n . Proof. (i) Using (A1) one has that g 1 (s) ≤ E[(π Z * 0 (s)) Z * 0 ], s ∈ [0, 1] and E[s Z n+1 | Z n = k] = E s L φ(k) P φ(k) i=1 f n,i , P φ(k) i=1 m n,i ≤ E s P φ(k) i=1 L φ(k) (f n,i ,m n,i ) = E s Y φ(k) φ(k) = π φ(k) (s) φ(k) , k ∈ Z + , n ∈ Z + 0 . (ii) Let φ(Z + ) = {φ(k), k ∈ Z + }.
g * n+1 (s) = j∈φ(Z + ) E s φ(L j ( P j i=1 f n,i , P j i=1 m n,i )) P (Z * n = j) ≥ j∈φ(Z + ) E s P j i=1 φ(L j (f n,i ,m n,i )) P (Z * n = j) = j∈φ(Z + ) π * j (s) j P (Z * n = j) = E (π * Z * n (s)) Z * n .
(iii) It is proved in a similar way that (ii) using the fact that φ is non-decreasing.

By Proposition 3.2, E[Z n+1 ] ≥ E[Z * n E[Y Z * n ]] and E[Z * n+1 ] ≤ (or ≥)E[Z * n E[Y * Z * n ]], n ∈ Z + if φ is non-increasing (or superadditive). Also, assuming that Y j ≥ Y 1 , j ∈ Z + 0 , one deduces that π j (s) ≤ π 1 (s), s ∈ [0, 1], j ∈ Z + 0 and, again from Proposition 3.2, g n+1 (s) ≤ g * n (π 1 (s)) and g * n+1 (s) ≥ (or ≤)g * n (π * 1 (s)) if φ is non-increasing (or superadditive), s ∈ [0, 1], n ∈ Z + . Thus, for n ∈ Z + , E[Z n+1 ] ≥ E[Z * n ]E[Y 1 ] and E[Z * n+1 ] ≤ (or ≥)N * (E[Y * 1 ]) n+1 ifφis non-increasing (or superadditive) (2) Let T n = n i=1 Z i , n ∈ Z + 0 and T * n = n i=0 Z * i , n ∈ Z + and let us denote by G n (s, t) = E[s Tn t Zn ], G * n (s, t) = E[s T * n t Z * n ] and H n (s, t) = E[s Tn t Z * n ] , s, t ∈ [0, 1], n ∈ Z + 0 .
Clearly G 1 (s, t) = g 1 (st) and

G * 0 (s, t) = (st) N * , s, t ∈ [0, 1]. Proposition 3.3 Assume (A1) and Y j ≥ Y 1 , j ∈ Z + 0 . For s, t ∈ [0, 1] (i) G n+1 (s, t) ≤ H n (s, π 1 (st)), n ∈ Z + 0 . (ii) If φ is non-increasing on (0, ∞) then G * n+1 (s, t) ≥ G * n (s, π * 1 (st)), n ∈ Z + . (iii) If φ is superadditive then G * n+1 (s, t) ≤ G * n (s, π * 1 (st)), n ∈ Z + .
Proof. (i) First note that, from (A1) and the fact that

Y j ≥ Y 1 , j ∈ Z + 0 , one deduces that E[s Z n+1 | F n ] ≤ (π 1 (s)) Z * n almost surely, where F n = σ(Z 1 , . . . , Z n ), n ∈ Z + 0 , hence G n+1 (s, t) = E s T n E[(st) Z n+1 | F n ] ≤ E s T n (π 1 (st)) Z * n .

A c c e p t e d m a n u s c r i p t

Controlled bisexual processes 7

(ii) Again by (A1) and taking into account that φ is non-increasing, one derives

that E[s Z * n+1 | F * n ] ≥ (π * 1 (s)) Z * n almost surely, where F * n = σ(Z * 0 , . . . , Z * n ), n ∈ Z + , hence G * n+1 (s, t) = E s T * n E[(st) Z * n+1 | F * n ] ≥ E s T * n π * 1 (st) Z * n .
(iii) It is proved in a similar manner that (ii) by using that φ is superadditive.

By Proposition 3.3(i), E[T n+1 ] ≥ E[Y 1 ] n i=1 E[Z * i ] + E[Z 1 ]
, n ∈ Z + 0 and, using Proposition 3.3(ii) and ( 2), assuming φ non-increasing, one deduces that

E[T * n+1 ] ≤ N * (n + 2)ifα = 1or ≤ N * (1 -α n+2 )(1 -α) -1 ifα = 1 (3) where α = E[Y * 1 ]. The reverse is obtained in (3) when φ is superadditive. Notice that, if N * = 1, then one has that Y 1 = Z 1 hence that π 1 (s) = g 1 (s), π * 1 (s) = g * 1 (s), s ∈ [0, 1], and α = E[φ(L 1 (f 0,1 , m 0,1 ))].
We now study some stochastic monotony properties about

{Z n } n≥1 , {Z * n } n≥0 , {F n } n≥1 and {M n } n≥1 .
Proposition 3.4 Assume (A2) and φ non-increasing on (0, ∞). Then for

k i ∈ Z + , j i ∈ φ(Z + ), i = 1, 2, with k 1 < k 2 , j 1 < j 2 , y ∈ R, and n ∈ Z + 0 , P (Z n+1 ≤ y | Z n = k 2 ) ≥ P (Z n+1 ≤ y | Z n = k 1 ) and P (Z * n+1 ≤ y | Z * n = j 2 ) ≥ P (Z * n+1 ≤ y | Z * n = j 1 ).
Proof. Using (A2), the fact that φ(k 2 ) ≤ φ(k 1 ), and that L φ(k1) is monotonic nondecreasing in each argument, one derives for y ∈ R and n ∈ Z On the other hand, for y ∈ R and n ∈ Z

+ P (Z n+1 > y | Z n = k 2 ) ≤ P   L φ(k1)   φ(k 2 ) i=1 f n,i , φ(k 2 ) i=1 m n,i   > y   ≤ P   L φ(k1)   φ(k 1 ) i=1 f n,i , φ(k 1 ) i=1 m n,i   > y   = P (Z n+1 > y | Z n = k 1 ).
+ P (Z * n+1 > y | Z * n = j 2 ) ≤ P φ L j 1 j 2 i=1 f n,i , j 2 i=1 m n,i > y ≤ P φ L j 1 j1 i=1 f n,i , j1 i=1 m n,i > y = P (Z * n+1 > y | Z * n = j 1 ).
It is easy to verify that similar results with the reverse can be established, along

the line of Proposition 3.4, if φ is non-decreasing. Let {(F (i) n , M (i) n )} n≥1 , i ∈ Z + 0 be independent versions of {(F n , M n )} n≥1 and let us denote by {Z (i) n } n≥1 and {Z * (i) n } n≥0
their associated sequences of originated and progenitor couples, respectively, with

Z * (i) 0 = 1, i ∈ Z + 0 .
Proposition 3.5 Assume (A1), (A2), and φ superadditive. Then, for k ∈ Z + and y ∈ R, P (Z * n+k ≤ y) ≤ P (

Z * k i=1 Z * (i) n ≤ y), n ∈ Z + and P (F n+k ≤ y) ≤ P ( Z * k i=1 F (i) n ≤ y) ( similarly P (M n+k ≤ y) ≤ P ( Z * k i=1 M (i) n ≤ y) ), n ∈ Z + 0 .
Proof. First, we shall prove that for n, k, j ∈ Z + and y ∈ R,

P (Z * k+n+1 ≤ y | Z * n+k = j) ≤ P   Z * k i=1 Z * (i) n+1 ≤ y Z * k i=1 Z * (i) n = j   (4)
In fact, using (A2), the superadditivity of φ and (A1), one obtains

P   Z * k i=1 Z * (i) n+1 ≤ y Z * k i=1 Z * (i) n = j   ≥ P   φ   LPZ * k i=1 Z * (i) n   Z * k i=1 Z * (i) n l=1 f (i) n,l , Z * k i=1 Z * (i) n l=1 m (i) n,l     ≤ y Z * k i=1 Z * (i) n = j   = P φ L j j l=1 f k+n,l , j l=1 m k+n,l ≤ y = P (Z * k+n+1 ≤ y | Z * n+k = j).
Note that for n = 0, using that Z * (i) 0

= 1, it is clear that P (Z * k ≤ y) = P (

Z * k i=1 Z * (i) 0 ≤ y). Assume that P (Z * k+n ≤ y) ≤ P ( Z * k i=1 Z * (i) n ≤ y). Then, considering that φ is
P (Z * k+n+1 ≤ y) ≤ ∞ j=0 P (Z * n+k+1 ≤ y | Z * n+k = j)P   Z * k i=1 Z * (i) n = j   ≤ ∞ j=0 P   Z * k i=1 Z * (i) n+1 ≤ y | Z * k i=1 Z * (i) n = j   P   Z * k i=1 Z * (i) n = j   = P   Z * k i=1 Z * (i) n+1 ≤ y   .
Using a similar reasoning are proved the other inequalities.

Remark 3.1 Under assumptions in Proposition 3.5, it can be also proved that:

(i) If φ(Z n ) ≤ Z n , n ∈ Z + 0 , i.e. emigration of couples is produced in each genera- tion, then P (Z n+k ≤ y) ≤ P ( Z * k i=1 Z * (i) n ≤ y), y ∈ R + , n ∈ Z + . (ii) If φ(Z n ) ≥ Z n , n ∈ Z + 0 , i.e. immigration of couples is produced in each gener- ation, then P (Z * n+k ≤ y) ≤ P ( Z k i=1 Z (i) n ≤ y), P (F n+k ≤ y) ≤ P ( Z k i=1 F (i) n ≤ y) and P (M n+k ≤ y) ≤ P ( Z k i=1 M (i) n ≤ y), y ∈ R + , n ∈ Z + .

Extinction probability

It is clear that the process (1) will become extinct if, in some generation, there are not any progenitor couples. Let us write by q N * = P (Z * n → 0 | Z * 0 = N * ) the extinction probability when the process is initiated with N * couples. Let us introduce the

sequences { Z n } n≥0 , {Z n } n≥0 and { Z n } n≥0 where Z n+1 = e Zn i=1 L 1 (f n,i , m n,i ), Z n+1 = Z n i=1 φ(L 1 (f n,i , m n,i )) and Z n+1 = φ( b Zn) i=1 L 1 (f n,i , m n,i ), n ∈ Z + , with Z 0 = Z 0 = N * ,
and

Z 0 = N 0 ≥ 1. We assume that N 0 is such that φ(N 0 ) ≤ N * . Note that { Z n } n≥0
and {Z n } n≥0 are asexual Galton-Watson processes with offspring distributions as the laws of Y 1 = L 1 (f 0,1 , m 0,1 ) and Y * 1 = φ(Y 1 ), respectively. On the other hand, (c) φ is superadditive and

sup k>0 k -1 φ(k) > E[Y 1 ] -1 .
Then q N * < 1, N * ∈ Z + 0 .

Proof. Suppose that condition (a) is satisfied. We shall prove, by induction on n, that Z * n ≥ Z n , n ∈ Z + . In fact, for n = 0, Z * 0 = Z 0 = N * ≥ 1. Assume that Z * n ≥ Z n , then, by (A1) and (A2),

Z * n+1 ≥ φ   Z * n i=1 L 1 (f n,i , m n,i )   ≥ φ   e Z n i=1 L 1 (f n,i , m n,i )   ≥ Z n+1 .
Thus q N * ≤ q N * = P ( Z n → 0| Z 0 = N * ) and, from Galton-Watson process theory, using the fact that E[Y 1 ] > 1, one deduces that q N * < 1 and therefore q N * < 1. We now assume condition (b) holds. It is sufficient to verify that Z * n ≥ Z n , n ∈ Z + .

Again by induction, it is clear that Z * 0 = Z 0 = N * . Suppose that Z * n ≥ Z n , then from (A1) and (A2),

Z * n+1 ≥ Z * n i=1 φ (L 1 (f n,i , m n,i )) ≥ Zn i=1 φ (L 1 (f n,i , m n,i )) Z n+1 .
Thus q N * ≤ q N * = P (Z n → 0 | Z 0 = N * ) and, taking into account that E[Y * 1 ] > 1, the result is derived. Finally, if (c) is satisfied then, considering that φ(0) = 0 and φ(k) ∈ Z + 0 , k ∈ Z + 0 , one obtains that q N * = P (Z n → 0 | Z * 0 = N * ). To complete the

  By (A1) and the fact that φ is non-increasing on (0, ∞), one obtains for s ∈ [0, 1] and n ∈ Z +

  , that for n, k ∈ Z + , y ∈ R fixed, {P (Z * k+n+1 ≤ y | Z * n+k = j)} j≥0 is anon-increasing sequence, the induction hypothesis, Lemma 1 (see Appendix) and (4), one deduces that

  et al. { Z n } n≥0 is an asexual controlled Galton-Watson process with offspring distribution as the law associated to Y 1 . Proposition 4.1 Assume (A1) and (A2). If any of the following conditions holds:(a) φ is non-decreasing, φ(k) ≥ k, k ∈ Z + 0 and E[Y 1 ] > 1. (b) φ is superadditive and E[Y * 1 ] > 1.

  Intuitively, (f n,i , m n,i ) represents the number of females and males descending from the ith couple of the nth generation. Hence (F n+1 , M n+1 ) is the number of females and males in the (n + 1)th generation, which form Z n+1 couples through the mating function L Z * n , where Z * n = φ(Z n ) denotes the number of progenitor couples, namely, the couples that after the control governed by φ really participated in the reproduction. We remark that, ifφ(Z n ) > Z n then φ(Z n ) -Z n couples are introduced in the population; if φ(Z n ) < Z n then Z n -φ(Z n ) couples leave the population; and no control is made if φ(Z n ) = Z n . It is easy to verify that {(Z n-1 , F n , M n )} n≥1 ,{Z n } n≥1 and {Z * n } n≥0 are homogeneous Markov chains with 0 being an absorbing state for {Z n } n≥1 and {Z * n } n≥0 .

	Controlled bisexual processes	3
	bers, respectively, and Z + 0 = Z Moreover its theoretical interest, the bisexual model introduced in this work has
	several practical implications in population dynamics. For example, in phenomena
	concerning to inhabit or re-inhabit environments with animal species which have sex-
	ual reproduction, the probable evolution of the numbers of females, males, originated
	couples, and progenitor couples, may be described in term of this model. Indeed, the
	motivation behind the process (1) is the interest in developing stochastic models to
	describe probabilistically such situations.

with Z + and R + denoting the non-negative integer and real num-

A c c e p t e d m a n u s c r i p t + -{0}.

Acknowledgement: We thank the anonymous referee for its constructive suggestions which have improved this paper.

Research supported by the Ministerio de Ciencia y Tecnología and the FEDER through the Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica, grants BFM2003-06074 and MTM2006-08891.

A c c e p t e d m a n u s c r i p t

Controlled bisexual processes 11 proof it is sufficient to verify that Z n ≥ Z n , n ∈ Z + 0 . Now, by induction on n, if n = 1, using (A1), (A2) and the fact that

Taking into account that φ is superadditive and therefore non-decreasing, one has that

, one deduces that q N * < 1 and consequently q N * < 1.

Remark 4.1 The concept of the mean growth rate per couple introduced in [START_REF] Bruss | A note on extinction criteria for bisexual Galton-Watson processes[END_REF] can be extended in a natural form to the class of processes (1). In fact, for each j ∈ φ(Z + 0 ) = φ(Z + ) -{0}, we define the mean growth rate per progenitor couple as the expected value

Note that, assuming (A1), (A2) and φ superadditive, using a similar reasoning to that one used in [START_REF] Molina | Bisexual Galton-Watson branching process with population-size dependent mating[END_REF], Theorem 3.2, one obtains that, if the offspring mean vector µ is a

(u i -u i+1 )s i + u n s n . Now, this inequality holds because u i -u i+1 ≥ 0, i = 1, . . . , n and u n ≥ 0.


Susanne Ditlevsen

A result on the first-passage time of an

Keywords: Laplace transform, Martingales, Eigenfunctions, Conditional moments, Hermite polynomials

Consider the first time an Ornstein-Uhlenbeck process starting from zero crosses a constant positive threshold. Assuming that the asymptotic mean is above the threshold, conditions on the asymptotic variance relative to the distance between the threshold and the asymptotic mean are given that ensures the finiteness of the positive Laplace transforms.

A c c e p t e d m a n u s c r i p t 1 Introduction

The first-passage time of an Ornstein-Uhlenbeck process through a constant threshold has been used in diverse areas of applied mathematics when it is reasonable to assume an underlying stochastic process in a problem of interest, which eventually will reach a certain level that leads to some observed event.

In biology, it has been used as a model for neuronal activity (see e.g. [START_REF] Bulsara | Cooperative behavior in periodically driven noisy integrate-and-fire models of neuronal dynamics[END_REF][START_REF] Lansky | On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity[END_REF][START_REF] Ricciardi | The Ornstein-Uhlenbeck process as a model of neuronal activity[END_REF][START_REF] Shimokawa | A firstpassage-time analysis of the periodically forced noisy leaky integrate-andfire model[END_REF]Tuckwell, Wan and Rospars, 2002), in survival analysis the model has been applied by [START_REF] Aalen | Survival models based on the Ornstein-Uhlenbeck process[END_REF], and also in mathematical finance it has found applications (see e.g. [START_REF] Jeanblanc | Modelling of default risk: An overview[END_REF][START_REF] Leblanc | Path dependent options on yields in the affine term structure[END_REF][START_REF] Linetsky | Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models[END_REF]. A large literature has been dedicated to find the distribution of the first-passage time (for an overview see [START_REF] Alili | Representations of the first hitting time density of an Ornstein-Uhlenbeck process[END_REF].

The theoretical results of the present paper have been used to define moment estimators from observations of first-passage times in the context of neuronal modeling [START_REF] Ditlevsen | Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model[END_REF]. The same techniques were applied in [START_REF] Ditlevsen | Estimation of the input parameters in the Feller neuronal model[END_REF] for defining estimators in the Cox-Ingersoll-Ross process [START_REF] Cox | A theory of the term structure of interest rates[END_REF], which in the neuronal literature is called the Feller process [START_REF] Lansky | On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity[END_REF] because [START_REF] Feller | Diffusion processes in genetics[END_REF] proposed it as a model for population growth.

In this paper a formula for the conditional moments of any order of the Ornstein-Uhlenbeck-process is first derived, and then used to define suitable martingales. The martingales are then applied to give conditions on the parameter space so that the positive Laplace transform of the first-passage time of the process through a constant threshold is finite.

The Ornstein-Uhlenbeck process

Consider the Ornstein-Uhlenbeck process:

dX t = -β(X t -α)dt + σdW t ; X 0 = x 0 = 0, ( 1
)
where W is a Wiener process, and θ = (α, β, σ) ∈ R × R + × R + . The generator of (1) is the differential operator

L θ = 1 2 σ 2 d 2 dx 2 -β(x -α) d dx

A c c e p t e d m a n u s c r i p t

defined for all twice differentiable functions. An eigenfunction ϕ(x; θ) for L θ with associated eigenvalue λ(θ) is a twice continuously differentiable function that fulfills

L θ ϕ(x; θ) = -λ(θ)ϕ(x; θ) (2)
for all x ∈ R. For simplicity consider the centered process Xt = X t -α, which has eigenfunctions

ϕ k = [ k 2 ] m=0 (-1) m k! (k -2m)!m! σ 2 4β m x k-2m
for k ∈ N, where [a] denotes the largest integer such that [a] ≤ a. The associated eigenvalues are λ k = kβ. This can easily be checked by substitution into (2). In Karlin and Taylor (1981, p.333) the eigenfunctions are given for

σ = β = 1. Note that ϕ k (x) = (σ 2 /4β) k/2 H k (x β/σ 2 )
, where H k are Hermite polynomials [START_REF] Lebedev | Special functions and their applications[END_REF]. For this model we have for t > s that [START_REF] Kessler | Estimating equations based on eigenfunctions for a discretely observed diffusion process[END_REF], where E[•] denotes expectation under the law of (1). This yields the formula for the conditional moments

E ϕ k ( Xt ; θ)| Xs = e -kβ(t-s) ϕ k ( Xs ; θ), (3)
E[ Xk t | Xs ] = e -kβ(t-s) [ k 2 ] m=0 k! (k -2m)!m! σ 2 4β m (e 2β(t-s) -1) m Xk-2m s . (4)
In effect, (4) is obviously true for k = 1 and 2. Assume it is true for k

-2m for all 1 ≤ m ≤ [ k 2 ]. First note that E ϕ k ( Xt ; θ)| Xs = [ k 2 ] m=0 (-1) m k! (k -2m)!m! σ 2 4β m E Xk-2m t | Xs = E Xk t | Xs + [ k 2 ] m=1 (-1) m k! (k -2m)!m! σ 2 4β m E Xk-2m t | Xs so that (3) yields E Xk t | Xs = e -kβ(t-s) [ k 2 ] m=0 (-1) m k! (k -2m)!m! σ 2 4β m Xk-2m s - [ k 2 ] m=1 (-1) m k! (k -2m)!m! σ 2 4β m E Xk-2m t | Xs .

A c c e p t e d m a n u s c r i p t

The last summand on the right hand side can by the induction assumption be written

- [ k 2 ] m=1 (-1) m k! (k -2m)!m! σ 2 4β m e -(k-2m)β(t-s) × [ k-2m 2 ] j=0 (k -2m)! (k -2(m + j))!j! σ 2 4β j (e 2β(t-s) -1) j Xk-2(m+j) s = - [ k 2 ] n=1 n m=1 (-1) m k! (k -2m)!m! σ 2 4β m e -(k-2m)β(t-s) × (k -2m)! (k -2n)!(n -m)! σ 2 4β (n-m) (e 2β(t-s) -1) (n-m) Xk-2n s = - [ k 2 ] n=1 k! (k -2n)!n! σ 2 4β n e -kβ(t-s) Xk-2n s × n m=1 n m (-e 2β(t-s) ) m (e 2β(t-s) -1) (n-m) = - [ k 2 ] n=1 k! (k -2n)!n! σ 2 4β n e -kβ(t-s) Xk-2n s (-1) n -(e 2β(t-s) -1) n
where we have used the identities N

n = n i=0 n i a (n-i) b i = a n + n i=1 n i a (n-i) b i
, where a = (e 2β(t-s) -1) and b = -e 2β(t-s) . This yields

E Xk t | Xs = e -kβ(t-s) [ k 2 ] m=0 (-1) m k! (k -2m)!m! σ 2 4β m Xk-2m s - [ k 2 ] n=1 k! (k -2n)!n! σ 2 4β n e -kβ(t-s) (-1) n -(e 2β(t-s) -1) n Xk-2n s so that the coefficient to Xk-2i s for 1 ≤ i ≤ [ k 2 ] is e -kβ(t-s) k! (k -2i)!i! σ 2 4β i (-1) i -(-1) i -(e 2β(t-s) -1) i = e -kβ(t-s) k! (k -2i)!i! σ 2 4β i (e 2β(t-s) -1) i
which is also the case for i = 0. We have finally obtained (4).

T = inf{t > 0; X t ≥ S > 0}, (5)
which is the first-passage time of X t through a constant threshold. The Laplace transform of T is given by the representation

E e λβT = exp{ α 2 β 2σ 2 }D λ α √ 2β/σ exp{ (α-S) 2 β 2σ 2 }D λ (α -S) √ 2β/σ = H λ α √ β/σ H λ (α -S) √ β/σ (6)
for λ < 0, where D λ (•) and H λ (•) are parabolic cylinder and Hermite functions, respectively, see Lebedev (1972, p.98) and Borodin and Salminen (2002, p.542). The result can be extended to λ > 0 with certain restrictions on the parameter space to ensure that E e λβT < ∞. A decreasing sequence of subsets of the parameter space is found by defining suitable martingales using the conditional moments of ( 1), which provides conditions on the parameters for which ( 6) is finite. The expression becomes particularly simple when k ∈ N, because then (6) reduces to a fraction of Hermite polynomials.

Main result

Theorem 1 Let X t and T be given by ( 1) and ( 5), respectively, and let k) is the largest root of the k'th Hermite polynomial. Then

(α, β, σ) = θ ∈ Θ (k) = {θ | α > S, σ 2 /β < (α -S)/λ (k) } for k ∈ N, where λ (
E e λβT = H λ α √ β/σ H λ (α -S) √ β/σ
for λ ≤ k, where H λ is the Hermite function.

Note first that this is intuitively true, since for λ a positive integer, ( 6) is well-defined when θ fulfills the condition given in Theorem 1.

Proof: Define the processes are martingales with respect to the natural filtration F t = σ(X s ; 0 ≤ s ≤ t), the sigma-algebra generated by X s for 0 ≤ s ≤ t. First observe that M (k) t have finite expectation for all k since they are polynomial functions of a Gaussian variable, which possesses moments of any order. Moreover, for s < t we have

M (k) t = [ k 2 ] m=0 (-1) m k! (k -2m)!m! σ 2 4β m (1 -e -2βt ) m e kβt Xk-2m t = h k t H k Xt e βt
E[M (k) t |F s ] = E[M (k) t | Xs ] = [ k 2 ] m=0 (-1) m k! (k -2m)!m! σ 2 4β m (1 -e -2βt ) m e kβt E[ Xk-2m t | Xs ] = [ k 2 ] m=0 [ k-2m 2 ] j=0 (-1) m k! (k-2(m+j))!m!j! σ 2 4β (m+j) (1-e -2βt
) m e kβs e 2mβ(t-s) (e 2β(t-s) -1) j Xk-2(m+j)

s = [ k 2 ] n=0 n m=0 (-1) m k! (k-2n)!m!(n-m)! σ 2 4β n (e 2β(t-s) -e -2βs ) m e kβs (e 2β(t-s) -1) n-m Xk-2n s = [ k 2 ] n=0 k! (k-2n)!n! σ 2 4β n e kβs Xk-2n s n m=0 n m (e -2βs -e 2β(t-s) ) m (e 2β(t-s) -1) n-m = [ k 2 ] n=0 (-1) n k! (k-2n)!n! σ 2 4β n e kβs Xk-2n s (1 -e -2βs ) n = M (k) s .
Therefore also

M (k)
T ∧t , the processes stopped at T , are martingales (Williams, 1991, p.99), where T is given by (5). This yields

α k = (-1) k E[M (k) 0 ] = (-1) k E[M (k) T ∧t ] = E h k T ∧t H k (α-X T ∧t )e β(T ∧t) 2h T ∧t . (7)
Define λ (k) to be the largest positive root of the kth Hermite polynomial. Then Szegö, 1975, p.46). The first four are

λ (1) < λ (2) < • • • < λ (k) < λ (k+1) < • • • (
λ (1) = 0, λ (2) = 1/ √ 2, λ (3) = 3/2 and λ (4) = (3 + √ 6)/2. Moreover, H k (x)
is positive and monotonically increasing for x > λ (k) . Define the decreasing sequence of subsets of the parameter space

Θ (k) = {θ | α > S, σ 2 /β < (α -S)/λ (k) }
where division by 0 is defined to be infinity. The first inequality in the definition of the subsets Θ (k) defines the supra-threshold regime, where the asymptotic mean α of X t is larger than the threshold S. The second inequality restricts the asymptotic standard deviation σ 2 /2β of X t to be smaller than a factor proportional to the distance between the asymptotic mean and the threshold, and inversely proportional to the largest positive root of the kth Hermite polynomial. Thus, when k increases, the variance has to be smaller for the exponential moments to be finite, which is a natural requirement. Assume θ ∈ Θ (k) , then

(α -X T ∧t )e β(T ∧t) 2h T ∧t = (α -X T ∧t )e β(T ∧t)
σ 2 /β (e 2β(T ∧t) -1)

> (α -S) σ 2 /β > λ (k)
for all t, so that if θ ∈ Θ (k) , then (7) yields

α k ≥ E h k T ∧t H k (α -S)e β(T ∧t) 2h T ∧t = [ k 2 ] m=0 g km (θ)E e kβ(T ∧t) (1 -e -2β(T ∧t) ) m = [ k 2 ] m=0 g km (θ) m i=0 m i (-1) i E e (k-2i)β(T ∧t) = [ k 2 ] m=0 g km (θ)E e kβ(T ∧t) + [ k 2 ] m=1 g km (θ) m i=1 m i (-1) i E e (k-2i)β(T ∧t)
where

g km (θ) = (-1) m k! (k -2m)! m! σ 2 4β m (α-S) k-2m . The coefficient to E e kβ(T ∧t) is [ k 2 ] m=0 g km (θ) = σ 2 4β k 2 H k (α -S) √ β σ
, which is positive when θ ∈ Θ (k) .

We can therefore rearrange

α k - [ k 2 ] m=1 g km (θ) m i=1 m i (-1) i E e (k-2i)β(T ∧t) [ k 2 ] m=0 g km (θ) ≥ E e kβ(T ∧t) (8)
This is valid for all k ∈ N. For k = 1 we obtain α/(α -S) ≥ E e β(T ∧t) for α > S, and when k = 2 then (8 ∧t) for α > S and σ 2 /2β < (α -S) 2 . Therefore, by induction and because Θ (k) ⊂ Θ (k-1) , the left hand side will be finite, say less than some constant K k depending on k. Taking limits on both sides we obtain We have thus obtained conditions on the parameter space for which E[e kβT ] < ∞, so that (6) can be applied. 2

) yields (α 2 -σ 2 /2β) / ((α -S) 2 -σ 2 /2β) ≥ E e 2β(T
K k ≥ lim t→∞ E e kβ(T ∧t) = E e kβT
Note that substituting E e (k-2i)β(T ∧t) in ( 8) with expression (6) for λ = k -2i we obtain that in the limit when t → ∞, (8) is an equality, using the same techniques as in the proof of (4). This implies that when θ ∈ Θ (k) , then

E[M (k) 0 ] = (-α) k = E[M (k) T ].
In Table 1 the first four moments are given with indication of the subset of the parameter space in which the expressions are valid.

2 1 √ 2 α 2 - σ 2 2β (α -S) 2 - σ 2 2β 3 3 2 α (α -S)     α 2 - 3σ 2 2β (α -S) 2 - 3σ 2 2β     4 3 + √ 6 2 α 2 - 3σ 2 2β 2 - 3σ 4 2β 2 (α -S) 2 - 3σ 2 2β 2 - 3σ 4 2β 2 Table 1
The first 4 moments of e βT , where T is the first-passage time of an Ornstein-Uhlenbeck process with parameters θ = (α, β, σ) through a constant threshold S, which is valid for θ ∈ Θ (k) = {θ | α > S, σ 2 /β < (α -S)/λ (k) }.

  m,n-m through the change of variable n = m + j, and (a + b)

  2

3

  The first-passage time through a constant threshold Define the stopping time

  where H k (•) are the Hermite polynomials and h t = σ 2 (e 2βt -1)/4β. The M (k) t

  -Uhlenbeck input current. Biol. Cybern., 86:137-145, 2002. D. Williams. Probability with Martingales. Cambridge University Press, 1991.

Acknowledgments

I am grateful to Michael Sørensen for helpful comments. Work supported by grants from the Danish Medical Research Council and the Lundbeck Foundation.