Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences

February 2005, Vol. 9, p. 38–73 DOI: 10.1051/ps:2005003

CONVERGENCE TO INFINITELY DIVISIBLE DISTRIBUTIONS WITH FINITE VARIANCE FOR SOME WEAKLY DEPENDENT SEQUENCES

J´ erˆ ome Dedecker

and Sana Louhichi

Abstract. We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the Gaussian and the purely non-Gaussian parts of the infinitely divisible limit. We also discuss the rate of Poisson convergence and emphasize the special case of Bernoulli random variables.

The proofs are mainly based on Lindeberg’s method.

Mathematics Subject Classification. 60E07, 60F05.

Received July 3, 2003.

Introduction

Infinitely divisible distributions play a fundamental role in the area of limit theorems of probability theory.

This role is described by the following result due to Khintchine: let (X_j,n)_1≤j≤n be a triangular array of i.i.d. random variables such thatX_0,n converges in probability to zero asn tends to infinity. Then the set of distribution that are limits (in the sense of weak convergence) of the distribution of _n

j=1X_j,ncoincides with the set of all infinitely divisible distribution (see for instance [25]).

A probability measure µ on IR is infinitely divisible if, for any positive integer n, there is a probability measureµ_n on IR such thatµ=µⁿ_n, whereνⁿ denotes then-fold convolution of the probability measureν with itself. Since 1930, we know that the the characteristic function ˆµ of any infinitely divisible distribution µ is given by the formula

ˆ

µ(z) = exp

−1

2σ²z²+iγz+ e^izx−1−izx1I_{|x|≤1}(x) ν(dx)

, for any z∈IR, (0.1)

Keywords and phrases.Infinitely divisible distributions, L´evy processes, weak dependence, association, binary random variables, number of exceedances.

1 Laboratoire de Statistique Th´eorique et Appliqu´ee, Universit´e Paris 6, Site Chevaleret, 13 rue Clisson, 75013 Paris, France;

[email protected]

2 Laboratoire de Probabilit´es, Statistique et mod´elisation, Universit´e de Paris-Sud, Bˆat. 425, 91405 Orsay Cedex, France;

[email protected]

c EDP Sciences, SMAI 2005

whereσ, γare real numbers andν is a positive measure on IR such that ν({0}) = 0 and

(x²∧1)ν(dx)<+∞.

This representation of ˆµby the triplet (σ², ν, γ) is unique. The formula (0.1) is known as the L´evy-Khintchine representation and (σ², ν, γ) as the generating triplet ofµ. The quantitiesσ²andν are called, respectively, the Gaussian varianceand theL´evy measureofµ.

If

|x|≥1x²ν(dx) is finite, then so is

x²µ(dx). If moreover

xµ(dx) = 0, then we have the representation by the couple (σ², ν)

ˆ

µ(z) = exp

−1

2σ²z²+ e^izx−1−izx ν(dx)

, for any z∈IR, (0.2)

which is equivalent to ˆ

µ(z) = exp e^izx−1−izx 1 x²dF(x)

, for any z∈IR, (0.3)

where F is the distribution function of some finite measure (i.e. F is bounded, nondecreasing, F(−∞) = 0 andF(∞) is finite). The distribution functionF is called Kolmogorov’s spectral function. Letµ¹_F denotes the probability measure with characteristic function given by (0.3) and define for any positive realtthe probability µ^t_F =µ¹_tF. The distributionµ^t_F has mean zero and variancetF(∞) and satisfies the equationµ^t_F∗µ^s_F =µ^t+s_F .

The connection betweenF and the generating couple (σ², ν) is given by F(x) =

_x

−∞t²ν(dt) +σ²1I_[0,+∞[(x).

ClearlyF(∞) =

t²ν(dt) +σ²=

x²µ(dx). The Gaussian varianceσ² is related toF via the relation

→0lim(F()−F(−)) =σ².

Hence the continuity ofF at zero ensures thatσ²= 0. In such a caseµis calledpurely non-Gaussian. Examples of centered infinitely divisible distributions µ with finite second moment are then given by the functionF or equivalently by the generating couple (σ², ν). TheGaussianmeasure is obtained when the L´evy measureν is zero i.e. when F = σ²1I_[0,+∞[. The infinitely divisible distribution µ is Poisson if σ² = 0, and ν = λδ₁ or equivalently when F =λ1I_[1,+∞[. More generally, theCoumpound Poisson measure is obtained when σ² = 0 and ν =λρ, where ρis a probability measure on IR such thatρ({0}) = 0. In that caseF(x) =λ_x

∞t²ρ(dt).

Other examples of infinitely divisible distributions on IR can be found in [28].

Let (X_j,n)_1≤j≤n be a triangular array of stationary centered random variables such thatE(X_0,n² ) converges to zero as ntends to infinity. Suppose that for eachn, (X_i,n)_1≤i≤n are i.i.d. From Theorem 2 of Chapter 4 in [12], we know thatS_n(t) =X_1,n+· · ·+X_[nt],nconverges in distribution toµ^t_F if and only if, for any continuity pointxofF,

n→∞lim nE(X_0,n² 1I_X0,n≤x) =F(x).

In this paper, our aim is to extend the result of Gnedenko and Kolmogorov to certain dependent sequences.

When it is relevant, we also study the convergence of the Donsker line {Sn(t), t∈[0,1]} to a L´evy process in the space of cadlag functions equipped with Skorohod’s distance. A stochastic process{X_t, t ≥0} on IR is a L´evy processif the following conditions are satisfied.

(1) For any choice ofn≥1 and 0≤t₀ < t₁< . . . < t_n, the random variables X_t0, X_t1−X_t0, X_t2−X_t1, . . .,X_tn−X_tn−1 are independent.

(2) X₀= 0 a.s.

(3) The distribution ofX_t+s−X_s does not depend ons.

(4) It is stochastically continuous.

(5) Almost surely,X_t(ω) is right-continuous int≥0 and has left limits int >0.

The only L´evy process with almost sure continuous paths is the Brownian motion while the only L´evy process with almost sure count paths is the Poisson process (a count path is a nondecreasing cadlag function which takes integers as values and has jumps of exactly 1 at its point of discontinuity).

Our paper is organized as follows. In Section 1 we study the problem of convergence to infinitely divisible distributions for sequences having an independent asymptotic representation (cf. Condition (1.1)). Under this condition convergence in distribution to µ_F is equivalent to

p→∞lim lim sup

n→+∞

pE

S_n²(1/p)1I_Sn(1/p)≤x

−F(x)= 0

(cf. Th. 1, Sect. 1). To prove this result we adapt Lindeberg’s method with increasing blocks in place of individual variables. The main idea is to split S_n(1) intopblocks distributed asS_n(1/p) and to replace them step by step by blocks of i.i.d. variables with lawµ^1/n_F .

Once the convergence ofS_n(1) toµ_F is guaranteed, it remains to describe the sequences constructed from the original one, that approximate the Gaussian and the purely non-Gaussian parts of the infinitely divisible lawµ_F. For this, we consider the truncated random variables

X_i,n(η) =f_η(X_i,n)−E(f_η(X_i,n)) and X_i,n^(η)=f^(η)(X_i,n)−IE(f^(η)(X_i,n)),

where the “decoupling” functionsf_η andf^(η)are defined byf_η(x) = (η∧x)∨(−η) andf^(η)(x) =x−f_η(x). In Propositions 1 and 2 we prove that, under a “Lindeberg-type condition” (cf. Condition (C₁) of Prop. 1), the small sum(i.e. the sum of the random variables which are small by truncation) is approximately normal with variance asymptotically equal to the jump of F at 0. While the essential sum(i.e. the sum of the essential parts of the variables) is approximated by a purely non-Gaussien distribution, provided that the quantity

pE







^[n/p]

i=1

X_i,n^(η)





2

1I_|[n/p]

i=1 X_i,n^(η)|≤





is sufficiently small (cf. Condition (C3) of Prop. 1). The total sum behaves as a convolution of the normal and the purely non-Gaussian distribution (cf. Prop. 2). For i.i.d. sequences this result is a consequence of the so-called “d´ecoupage de L´evy” method (see for instance Lemma 3-1 in [1]). The key for the proofs of Propositions 1 and 2 is Lemma 4 which reduces the behaviour of the total sum to that of the essential sum, as soon as the small variables obey a “Lindeberg-type condition”.

In Section 2, we apply the results of Section 1 to associated random variables. This notion of positive dependence comes independently from physics (mostly through the FKG inequality, cf. [11]), reliability and statistics [3, 10]. In Theorem 2 we give sufficient conditions on the individual random variables (or at least on some finite sums of the random variables) for the finite dimensional distributions ofS_n(t) to converge to those of a L´evy process. More precisely the main conditions are: the sequences VarS_n(1) and VarS_n^(η)(1) converge respectively to some positive constantsF(∞) andG(∞), and for any positive integerN

η→0limlim sup

n→+∞n E

S_N,n^(η)21I_S(η) N,n≤x

−E

S_N^(η)2_−1,n1I_S(η) N−1,n≤x

=F_N(x), (0.4)

where F_N is a bounded variation (BV) function. If soF_N converges weakly to the distribution function Gof some finite measure. The sum S_n(1) is then approximated by the convolution of the normal law with variance

F(∞)−G(∞) andµ_G (see Th. 2 for more details). In Section 2.2, we study the convergence to L´evy processes (cf. Th. 3 and Prop. 3). We then deduce sufficient conditions for the convergence to Wiener or Poisson processes (cf. Cors. 3 and 4). In Section 2.3, we consider the special case of Bernoulli distributed random variables. Let (Y_i,n) be an array of associated and Bernoulli distributed random variables with parameterp_n such that np_n is bounded. We suppose that the variance of the corresponding sum S_n,n(Y) converges and that (0.4) holds for S_N,n(Y). We prove that the limiting law is integer-valued coumpound poisson. When Cov(X_0,n, X_i,n) = 0 fori≥m+ 1, the conditions that we impose are comparable to the necessary and sufficient conditions obtained by [15] form-dependent Bernoulli-distributed arrays (cf. Cor. 5 and the remarks below for more details). Finally, we give a bound for the Dudley distance between a sum of Bernoulli random variables and the Poisson distribution by using Lindeberg’s method once more (see the proof of Prop. 4). Those results apply to stationary and Gaussian sequences with positive correlation function and provide upper bounds for the number of exceedances close to that obtained by [13].

In Section 3, we give the proofs for weakly dependent sequences. In particular, we give useful bounds for Conditions (C1) and (C3) (cf. Lems. 5 and 6 respectively). In Section 4, we give the proofs for associated sequences. The rates for the convergence to the Poisson distribution are proved in Section 5 (cf. Th. 4).

1. Weak convergence to infinitely divisible distribution for some dependent sequences

In this section we give an analogue of Theorem 1 in [8] for some sequences satisfying other type of dependence.

More precisely, our dependence assumptions are comparable to Condition B in [16] and [17] (cf. Condition (1.1) below). This condition suggests an asymptotic independent representation of the sum S_n(1) by blocs of inde- pendent random variables: there exists an i.i.d. sequence ( ˜U_i,n) with marginal distributionS_n(1/p_n), such that the sum ˜S_n=_pn

i=1U˜_i,n andS_n(1) have the same limiting behaviour under suitable conditions onp_n.

Theorem 1. Let (X_i,n)_i∈N,n∈N be a stationary array of square integrable and centered real-valued random variables and define S_n(t) = X_1,n+· · ·+X_[nt],n. Suppose that E(X_0,n² ) tends to zero as n tends to infinity.

Assume that, for any z∈IRand anyt∈[0,1],

p→+∞lim lim sup

n→+∞

Eexp(izS_n(t))−

Eexp(izS_[nt](1/p))_p= 0. (1.1) Let F be the distribution function of some finite measure. The following statements are equivalent:

(i) For any t∈[0,1], the sequence (S_n(t))_n≥0 converges in distribution to the probability µ^t_F. (ii) For any continuity point x(including+∞) of F,

p→∞lim lim sup

n→+∞

pE

S_n²(1/p)1I_Sn(1/p)≤x

−F(x)= 0. (1.2)

Note that the nature of Condition (1.2) is very close to that of condition LD in [17]. In both cases, the idea is to calculate the parameters of the limiting distribution on the base of properties of distributions of finite sums.

Remark. LetS_m,n=X_1,n+· · ·+X_m,nand define the setE of all positive integersmsuch that

∀z∈IR, ∀t∈[0,1], lim

n→+∞

Eexp(izS_n(t))−(Eexp(izS_m,n))^[[nt]/m]= 0. (1.3) Let m₀(X) be an element of the set E, with the convention that m₀(X) = +∞ if E = ∅. The term lim_m→m0(X)f(m) will designate either f(m₀(X)) if m₀(X) is finite, or the usual limit if m₀(X) = ∞. In- stead of (1.1), we can suppose that the sumS_n(t) admits an asymptotic independent representation by a sum of i.i.d. blocs with lengthmindependent ofn:

m→mlim0(X)lim sup

n→+∞

Eexp(izS_n(t))−(Eexp(izS_m,n))^[[nt]/m]= 0. (1.4)

If we assume moreover, instead of (1.2) that there exists a distribution functionF such that for any continuity pointx(including +∞) ofF,

m→mlim0(X)lim sup

n→+∞

n mE

S_m,n² 1I_Sm,n≤x

−F(x)= 0, (1.5)

then the conclusion of Theorem 1 still holds. An interesting case arises when 1 belongs toE. In such a case the sequence behaves as its independent version, as shown in the following corollary.

Corollary 1. Let(X_i,n)be as in Theorem 1. Suppose that for anyz∈IR, and any t∈[0,1]

n→+∞lim

Eexp(izS_n(t))−(Eexp(izX_0,n))^[nt]= 0. (1.6) Let F be a given distribution function. Then the statement (i) of Theorem 1 is equivalent to the following: For any continuity point x(including+∞) of F,

n→+∞lim nE

X_0,n² 1I_X0,n≤x

=F(x). (1.7)

We shall justify this remark in the proof of Theorem 1.

We now discuss an equivalent statement for (1.2). For η > 0, consider the two 1-Lipschitz functions f_η andf^(η) defined byf_η(x) = (η∧x)∨(−η) andf^(η)(x) =x−f_η(x). Set

X_i,n(η) =f_η(X_i,n)−E(f_η(X_i,n)) and X_i,n^(η)=X_i,n−X_i,n(η) and define for anyt∈[0,1]

S_n^(η)(t) = [nt]

i=1

X_i,n^(η) and S_n,η(t) = [nt]

i=1

X_i,n(η).

ClearlyS_n(t) =S_n^(η)(t) +S_n,η(t). For sufficiently small η, the random variableX_i,n(η) represents small values, while X_i,n^(η) is the essential part of X_i,n. Under a “Lindeberg-type condition” on the small sum S_n,η(t), the behaviour of the original sumS_n(t) is described by its essential partS_n^(η)(t) as shown in the following proposition.

Proposition 1. Let(X_i,n)be a stationary array of square integrable and centered real-valued random variables.

Consider the three conditions (C1) For any >0 lim

η→0lim sup

p→+∞ lim sup

n→+∞pE

S_n,η² (1/p)1I_{|Sn,η(1/p)|≥}

= 0.

(C2) The four sequences (nEX_0,n² ),(V_n(p)) = (pVarS_n(1/p)), (V_n,η(p)) = (pVarS_n,η(1/p))and (V_n^(η)(p)) = (pVarS_n^(η)(1/p))are uniformly bounded overn,pandη.

(C₃) lim

→0lim sup

η→0 lim sup

p→+∞ lim sup

n→+∞pE(S^(η)2_n (1/p)1I_−<S(η)

n (1/p)≤) = 0.

If C₁,C₂ andC₃ hold, then (1.2) is equivalent to: for any continuity point xofF

η→0limlim sup

p→+∞ lim sup

n→+∞

pE

S_n^(η)2(1/p)1I_S(η) n (1/p)≤x

−F(x) + 1I_x≥0

V_n(p)−V_n^(η)(p)= 0. (1.8) In particular, lettingσ²= lim

→0(F()−F(−)), then(C1), (C2),(C3)and (1.2) ensure that

η→0limlim sup

p→+∞ lim sup

n→+∞

V_n(p)−V_n^(η)(p)−σ²= 0. (1.9)

In Sections 3.4 and 3.5 we give useful inequalities in order to check Conditions (C1) and (C3). We suppose now that Conditions (1.1), (C1), (C2), (C3) and (1.2) are all satisfied. Our main task in the following proposition is to describe the sequences constructed from (X_i,n) that approximate respectively the Gaussian part and the purely non-Gaussian part. Let ( ˜U_i,n), ( ˜U_i,n(η)), ( ˜U_i,n^(η)) be three sequences of i.i.d. r.v’s distributed asS_n(1/p), S_n,η(1/p) andS_n^(η)(1/p), respectively. Define

S˜_n:=

p i=1

U˜_i,n, S˜_n^(η):=

p i=1

U˜_i,n^(η), S˜_n,η:=

p i=1

U˜_i,n(η).

The following proposition shows that for small η, the distribution L(S_n(1)) of the partial sum S_n(1) is well approximated by the convolution of L( ˜S_n^(η)) and L( ˜S_n,η). The two distributions L( ˜S_n^(η)) and L( ˜S_n,η) are respectively approximated by a purely non-Gaussian distribution and a Gaussian distribution.

Proposition 2. Let (X_i,n)_i∈N,n∈N be as in Theorem 1 and define the functionsG^(η)_n,p by the equality G^(η)_n,p(x) = F(x)−1I_x≥0(V_n(p)−Vn^(η)(p)). If (1.1),(C1),(C2)and(C3)are all satisfied, then the statement (i) of Theorem 1 (with t= 1) is equivalent to:

η→0limlim sup

p→+∞ lim sup

n→+∞

E

exp(izS˜_n,η)

−exp(−z²

2 V_n,η(p))

= 0. (1.10)

η→0limlim sup

p→+∞ lim sup

n→+∞

E

exp(izS˜^(η)_n ) −exp

e^izx−1−izx

x² dG^(η)_n,p(x)

= 0. (1.11)

η→0limlim sup

p→+∞ lim sup

n→+∞ pCov

S_n^(η)(1/p), S_n,η(1/p)

= 0. (1.12)

η→0limlim sup

p→+∞ lim sup

n→+∞

E(exp(izS_n)−E

exp(izS˜_n,η) E

exp(izS˜_n^(η))= 0. (1.13) Remark. Let G(x) = F(x)−1I_x≥0(F(0⁺)−F(0⁻)). This function inherits all the properties of F and is continuous at 0. To summarize, Conditions (1.1), (C₁), (C₂), (C₃) and (i) of Theorem 1 prove that the distribution of ( ˜S^(η)_n ) is approximated by the purely non-Gaussian L´evy distribution µ_G while the distribution of ( ˜S_n,η) is described by the Gaussian distribution with varianceF(0⁺)−F(0⁻) (also equals toF(+∞)−G(+∞)).

2. Convergence to L´ evy processes for associated variables

We now apply the results of the previous section to stationary arrays fulfilling a condition of positive depen- dence called association. We also discuss convergence to L´evy processes for such sequences.

Let (X_i,n)_i∈N,n∈Nbe as in Theorem 1. It is an array ofassociatedrandom variables if for everyn:

Cov(h(X_i,n, i∈A), k(X_i,n, i∈B))≥0,

wherehandkare coordinatewise nondecreasing real-valued functions andA,Bare finite subsets ofN(we refer to [3, 10] and the references therein for more details on associated random variables).

Up to our knowledge, the convergence to infinitely divisible distributions for arrays of associated variables is studied separately in the case of Gaussian or Poisson distributions (cf. [22] for an overview). For a sta- tionary sequence (X_i) of square integrable random variables, the Gaussian limits are obtained as soon as the Cox-Grimmett coefficient (cf. [7])

U(n) := 2

+∞

j=n+1

Cov(X₁, X_j)

tends to zero asntends to infinity. Poisson limits, or more generally convergence results for triangular arrays of independent variables, are extended to associated variables provided that

n→+∞lim

1≤j<k≤n

Cov(X_j,nX_k,n) = 0, (2.1)

(cf. Prop. 1 in [23]). In order to get a more general result containing both Gaussian and Poisson limits, we generalize the Cox-Grimmett coefficient: let

R_a(N) = lim sup

n→+∞ n

n r=N

Cov(X_0,n, X_r,n).

LetN_a,0(X) = inf{N, R_a(N) = 0}(N_a,0(X) may be infinite). The main assumptions are

N→Nlima,0(X)lim sup

n→+∞ n

n r=N

Cov(X_0,n, X_r,n) = 0 and (2.2)

N→Nlima,0(X)lim sup

n→+∞

nE(X_0,n² ) + 2n

N−1

r=1

Cov(X_0,n, X_r,n)−F(∞)

= 0, (2.3)

where F(∞) is some nonnegative number. For stationary arrays of associated variables, Condition (2.2) with N_a,0(X) = 1 is exactly (2.1).

2.1. Convergence of the finite dimensional distributions

B(N): There exists a BV functionF_N such that for any continuity pointxof F_N,

η→0limlim sup

n→+∞n E

S_N,n^(η)21I_S(η) N,n≤x

−E

S_N^(η)2_−1,n1I_S(η) N−1,n≤x

=F_N(x).

As for (X_i,n), define

R^(a)(N) = lim sup

η→0 lim sup

n→+∞ n

n r=N

Cov

X_0,n^(η), X_r,n^(η)

. (2.4)

andN₀^(a)(X) = inf{N, R^(a)(N) = 0}. We make the same assumptions on (X_i,n^(η))_ias we make on (X_i,n), that is lim

N→N₀^(a)(X)lim sup

η→0 lim sup

n→+∞ n

n r=N

Cov

X_0,n^(η), X_r,n^(η)

= 0 and (2.5)

lim

N→N₀^(a)(X)lim sup

η→0 lim sup

n→+∞

nE

X_0,n^(η)2

+ 2n

N−1

r=1

Cov

X_0,n^(η), X_r,n^(η)

−G(∞)

= 0, (2.6)

whereG(∞) is some positive real number.

Theorem 2. Let (X_i,n)_i∈N,n∈N be a stationary array of associated square integrable and centered real-valued random variables. Suppose thatE(X_0,n² )tends to zero asntends to infinity. Assume that Conditions (2.2), (2.3), (2.5) and (2.6) hold. Assume moreover that there exists a nondecreasing sequence of positive integers (N_i)_i∈N^∗ converging toN₀^(a)(X)such thatB(N_i)holds for anyiinN^∗. Then there exists a distribution functionGsuch that, for any bounded and three times continuously differentiable functiongwith compactly supported derivatives

lim

Ni→N₀^(a)

gdF_Ni =

gdG. (2.7)

If moreover G is continuous at 0, then the finite dimensional distributions of the process {S_n(t), t ∈ [0,1]} converges in distribution to those of the L´evy process of law µ_F, whereF is given by

F(x) =G(x) + 1I_0<x(F(∞)−G(∞)). Recall that F(∞)andG(∞) have been defined in (2.3) and (2.6).

Remark. Suppose that, for each fixed N, the BV function F_N, given in B(N) is of the form F_N(x) = _N

j=1λ_j,N1I_j<x,where (λ_j,N) is a sequence of real numbers. Suppose moreover that the requirements of Theorem 2 are satisfied. Then the distribution functionG, whose existence is guaranteed by Theorem 2 satisfiesG(x) = N₀^(a)(X)

j=1 λ_j1I_j<x, where the positive numbersλ_j are the limit ofλ_j,N. In this case the limit law is given by its characteristic function

ˆ

µ(z) = exp



−(F(∞)−G(∞))z² 2 +

N₀^(a)(X) j=1

λ_j(e^ijz−1−ijz)1 j²



·

This law is the convolution of some Gaussian distribution with some integer-valued Coumpound Poisson distribution.

2.2. Convergence to L´ evy processes

Let D([0,1]) be the space of cadlag functions equipped with Skorohod’s distance d. Convergence in the Skorohod topology is somewhat restrictive: for instance if x_n = 1I_[1/2,1] and y_n = 1I[1/2−1/n,1], both sequences converge in (D([0,1]), d) but x_n +y_n is not even relatively compact in that space. It means that for the convergence in (D([0,1], d)), two jumps of given size cannot become closer and closer. Hence, to obtain the weak convergence of{S_n(t), t∈[0,1]}in (D([0,1]), d), it is necessary to impose additional conditions on (X_i,n) (see Rem. 6 in [8] for more details). Since the sample paths of the Wiener process are continuous, it is natural to make these assumptions on the array (X_i,n^(η)) only.

Theorem 3. Let (X_i,n)_i∈N,n∈N be a stationary array of associated square integrable and centered real-valued random variables. Suppose that E(X_0,n² ) tends to zero as n tends to infinity. Assume moreover that Condi- tions (2.2) and (2.3) hold and that

η→0limlim sup

n→+∞n n r=1

Cov

X_0,n^(η) , X_r,n^(η)

= 0. (2.8)

Suppose that there exists a distribution functionG, continuous at0, such that for any point of continuityxofG

η→0limlim sup

n→+∞nE

X_0,n^(η)21I_X(η) 0,n≤x

=G(x).

Then the sequence {Sn(t), t ∈ [0,1]} converges in distribution to the L´evy process of law µ_F in the space (D([0,1]), d). The functionF is given by F(x) =G(x) + 1I_x≥0(F(∞)−G(∞)).

Remark. Conditions (2.2), (2.3) and (2.8) imply the tightness of the sequence of process{Sn(t), t∈[0,1]}in the space (D([0,1]), d).

Remark. If N_a,0(X) = 1, thenN₀^(a)(X) = 1 (cf. Sect. 4, the inequality (4.6) for a justification). In this case, the sumS_n(1) admits an asymptotic independent representation by blocs of length 1. In other words, if N_a,0(X) = 1 then the setE defined in (1.3) is nonempty and equals toN^∗ (we refer the reader to Cor. 2 below and to its proof for a justification of this remark), hence one may expect a limit theorem under conditions close to that obtained for independent sequences. Those ideas were first mentioned by [23] in order to prove the convergence in distribution ofS_n(1).

Corollary 2. Let(X_i,n)_i∈N,n∈N be as an in Theorem 3. Suppose that

n→+∞lim n n r=1

Cov(X_0,n, X_r,n) = 0. (2.9)

Let F be the distribution function of some finite measure. Then the following statements are equivalent (i) For any continuity point xofF (including +∞)

n→+∞lim nE

X_0,n² 1I_X0,n≤x

=F(x). (2.10)

(ii) The sequence of process {S_n(t), t ∈ [0,1]} converges in distribution in D([0,1]) to the L´evy process of law µ_F.

Theorem 3 contains both the Gaussian and the Poisson limits as shown in the following proposition (we omit the proof, which follows immediately from Th. 3).

Proposition 3. Let (X_i,n)_i∈N,n∈N be as an in Theorem 3. Assume that Conditions (2.2), (2.3) and (2.8) are satisfied. Suppose moreover that there exists a positive real numberλsuch that

η→0limlim sup

n→+∞nE

X_0,n^(η)21I_X(η) 0,n≤x

=λ1I_1≤x, (2.11)

for anyx= 1, including+∞. Then the sequence of processes{S_n(t), t∈[0,1]}converges in distribution in the space(D([0,1]), d)to the L´evy process of lawµ_F. The characteristic function µˆ of µ_F is given by

ˆ

µ(z) = exp

−1

2σ²z²+λ(e^iz−1−iz)

,

whereσ²=F(∞)−λandF(∞)is the positive constant defined in (2.3).

Remark. Clearlyσ² fulfills

σ²= lim

η→0lim sup

n→+∞

VarS_n(1)−nE X_0,n^(η)2

. (2.12)

2.2.1. Convergence to Wiener processes

Corollary 3. Let (X_i,n)_i∈N,n∈N be a stationary array of associated square-integrable and centered random variables. Suppose that E(X_0,n² ) tends to zero as n tends to infinity and that Conditions (2.2) and (2.3) are satisfied withF(∞) = 1. If moreover

n→+∞lim nE

X_0,n² 1I_|X0,n|≥

= 0 f or all >0, (2.13)

then{S_n(t), t∈[0,1]} converges in distribution in(D([0,1]), d)to the standard Wiener process.

For stationary and centered associated sequence with_∞

r=1Cov(X₀, X_r)<+∞,Corollary 3 applied to the arraysX_i,n=X_i/√

VarS_n leads to the CLT theorem already proved in [24].

2.2.2. Convergence to Poisson processes

Corollary 4. Let (X_i,n)_i∈N,n∈N be a stationary array of associated square-integrable and centered random variables. Assume that E(X_0,n² )tends to zero as ntends to infinity and consider the limits

n→+∞lim nE

X_0,n² (1∧ |X_0,n−1|)

= 0 and lim

n→+∞nE(X_0,n² ) =λ. (2.14)

1. If (2.2), (2.8) and (2.14) hold, then (2.9) holds.

2. Suppose moreover that Condition (2.9) holds. Then, the process {S_n(t), t∈[0,1]} converges in distri- bution in (D([0,1]), d) to the centered Poisson process {P(λt)−λt, t ∈ [0,1]} (i.e. the L´evy process with distribution functionF =λ1I_[1,+∞[) if and only if (2.14) holds.

The convergence in distribution ofS_n(1) to a Poisson limit was established in [23], under the Conditions (2.9) and (2.14).

2.3. On the convergence of Bernoulli random variables

Let (Y_i,n) be an array of Bernoulli distributed random variables with parameterp_n such thatnp_n is bounded and define the centered triangular arrayX_i,n byX_i,n=Y_i,n−p_n. We are interested in the asymptotic behavior of the processS_n(Y, t) =S_n(t) + [nt]p_n=Y_1,n+· · ·+Y_[nt],n, in the context of association.

Forη sufficiently small, we can choosenlarge enough such that

f^(η)(−p_n) = 0, f^(η)(1−p_n) = 1−p_n−η, f^(η)(X_0,n) = (1−p_n−η)Y_0,n. (2.15) We first deduce from those estimations thatN_a,0(X) =N₀^(a)(X) =N₀(Y).Let us now describe the conditions on the sequence (Y_i,n) under which the requirements of Theorem 2 are satisfied. From (2.15), we infer that (2.2), (2.3), (2.5) and (2.6) are satisfied as soon as

N→Nlim0(Y)lim sup

n→+∞

nVar(Y_0,n) + 2n

N−1 r=1

Cov(Y_0,n, Y_r,n)−F(∞)

= 0 (2.16)

and lim

N→N0(Y)lim sup

n→+∞n n r=N

Cov(Y_0,n, Y_r,n) = 0. (2.17)

Suppose moreover that, there exists a BV functionF_N such that for anyx, point of continuity ofF_N,

n→∞lim nE

S²_N,n(Y,1)1I_{SN,n(Y,1)≤x}

−nE

S_N−1,n² (Y,1)1I_{SN−1,n(Y,1)≤x}

=F_N(x), (2.18) then some elementary estimations based on (2.15) show that ConditionB(N) holds. Moreover 0 is a continuity point ofF_N sinceS_N,n(Y,1) is integer-valued. Combining these remarks with Theorem 2, we obtain the following corollary.

Corollary 5. Let(Y_i,n)be an array of Bernoulli-distributed random variables with parameterp_n such thatnp_n is bounded. Assume that (2.16) and (2.17) are satisfied, and that there exists a nondecreasing sequence of positive integers (N_i)_i∈N∗ converging to N₀(Y)such that (2.18) holds for any N_i. Then the finite dimensional distributions of the process{S_n(t), t∈[0,1]}converges in distribution to those of the L´evy process of the purely non-Gaussian law µ_G, whereGis the weak limit of (F_Ni)in the sense of (2.7) andG(∞) =F(∞).

Remark. Assume thatnp_n converges to some positive constantλ. Then under the assumptions of Corollary 5, S_n(Y, t) converges to X_t+λwhere X_thas law µ^t_G. From (2.18), it is clear that Gis piecewise constant with jumps at integer points, so that the limiting distribution of S_n(Y, t) is necessarily integer-valued coumpound Poisson.

Remark. Form-dependent arrays, necessary and sufficient conditions for the convergence ofS_n(Y,1) are given in [15]. Using our notations, these conditions are equivalent to: there exists a distribution functionF_m+1 such that

n→∞lim nE(S_m+1,n² (Y,1)1I_{Sm+1,n(Y,1)≤x})−nE(S_m,n² (Y,1)1I_{Sm,n(Y,1)≤x}) =F_m+1(x). (2.19) If N₀(Y) =m+ 1, we obtain the same condition as in the m-dependent case. In that case the distribution of the variable S_n(Y,1) is the law of the sum V₁+· · ·+V_N where N is Poisson distributed with parameter

λ=_m+1

i=1 (F_m+1(i)−F_m+1(i−1))/i²and is independent of the sequence (V_k)_k≥1which is i.i.d. with marginal distributionP(V1=i) = (F_m+1(i)−F_m+1(i−1))/(λi²). See Theorem 1.1 in [20] for more details.

Remark. Under the conditions of Corollary 5, we can establish the joint convergence of (S_n(t₁), . . . , S_n(t_k)) for any k-tuple (t₁, . . . , t_k) (cf. Sect. 4.2 and Lem. 15). Therefore, there is also convergence for the point process S_n(Y, A) =

i∈nAY_i,n indexed by Borel subsets of [0,1] (see Th. 4.2 in [19]).

We now discuss the number of exceedances under association (note that the method of the proof can be extended to other dependent variables). We refer to [14] for the importance of the exceedance process in extreme value theory. Let (X_i)_1≤i≤n be a collection of random variables and letS_n counts for how manyi’sX_i exceeds z_i, where the levels (z_i)_1≤i≤n are any real numbers. A poisson approximation for the law of S_n is appropriate whenever the probabilities of exceedance are small, as shown in the following proposition.

Proposition 4. Let(X_i)_i∈Nbe a stationary sequence of associated real-valued random variables, andB₁³(IR)be the set of three-times continuously differentiable real-valued functions for which h∞ ≤1 and h⁽³⁾∞ ≤1.

Let (z_i)_i∈Nbe real numbers and define p_n=P(X₁> z_n)andλ_n=np_n. Then

sup

h∈B₁³(IR)

Eh

_n

i=1

(1I_Xi>zn−p_n)

−µ_1,λn(h)

≤9np²_n+ 2n

n−1

r=1

Cov (1I_X0>zn,1I_Xr>zn),

whereµ_1,λn denotes the centered Poisson law P(λn)−λ_n.

Proposition 4 is based on Lindeberg’s method. Poisson approximation for dependent trials was discussed by Chen since 1975 (cf. [6]). Chen’s method is the adaptation to the Poisson distribution of Stein’s differential method for the normal distribution (cf. [29]). We refer to the monograph [2] Barbour for the Stein-Chen method as well as for the wide applicability of the Poisson approximations.

Let (X_i)_i∈Nbe a standardized stationary associated normal sequence, which means that the functionr(k) = Cov(X₀, X_k) is always positive (see [26]). Let (z_n) and λ_n be real numbers such that λ_n = n(1−Φ(z_n)), where Φ is the distribution function of the standard normal law. Suppose that sup_nλ_n<∞. An estimation of Cov

1I_Xi>zn,1I_Xj>zn as it is done in [13] together with Proposition 4 leads to the following: Ifr(k)≥0 for eachkandr(k)≤C/lnkfork≥2, then

sup

h∈B₁³(IR)

Eh

_n

i=1

(1I_Xi>zn−p_n)

−µ_1,λn(h) =O

n−(1+ρ)/(1+ρ)(lnn)^−ρ/(1+ρ)+lnn n

n k=1

|r(k)|

,

where ρ= max(r(1), r(2), . . .). Similar results are contained in [13], but their approach is different from ours (see also [21]).

3. Proofs for weakly dependent sequences 3.1. Proofs of Theorem 1 and Corollary 1

We first recall a technical result which is the main tool for the proof of Theorem 1.

Lemma 1. Let (Y_i,n)_i∈N,n∈N and (_i,n)_i∈N,n∈N be two independent arrays of i.i.d and centered real valued random variables, such that lim_n→+∞E(Y_0,n² ) = lim_n→+∞E(²_0,n) = 0. Let K_n be a positive integer. Let x₀≤. . .≤x_N be a finite grid and definef_j(t) =t²1I_xj<t≤xj+1. Then, for any functionhthree times differentiable

with h∞<∞,h⁽³⁾∞<∞, we have

E

h _K

n

i=1

Y_i,n

−h _K

n

i=1

_i,n

≤ h∞K_n

N−1

j=0

|E(f_j(Y_1,n))−E(f_j(_1,n))|

+h∞K_nE

Y_1,n² 1I_{Y1,n∈]x0,xN]}

+h∞K_nE

²_1,n1I_{1,n∈]x0,xN]} +K_n

N−1

j=0

E

f_j(Y_1,n)

h∞∧ h⁽³⁾∞|Y1,n−x_j|

+K_n

N−1

j=0

E f_j(_1,n)

h_∞∧ h⁽³⁾_∞|_1,n−x_j|

. (3.1)

Proof of Lemma 1. Letx₀≤. . .≤x_N beN+ 1 fixed real numbers. Letgbe a bounded function with bounded first derivative. Clearly

E(Y_i,n² g(Y_i,n)) =

N−1 j=0

E

Y_i,n² (g(Y_i,n)−g(x_j)) 1I_{xj<Yi,n≤xj+1} +

N−1 j=0

E

Y_i,n² g(x_j)1I_{xj<Yi,n≤xj+1} +E

Y_i,n² g(Y_i,n)1I_{Yi,n∈]x0,xN]} .

According to the properties of the functiong, we deduce from the last equality together with the analogous one forE(²_i,ng(_i,n)) that

E(Y_i,n² g(Y_i,n))−E(²_i,ng(_i,n))≤ g∞ N−1

j=0

|E(fj(Y_i,n))−E(fj(_i,n))|

+g∞E(Y_i,n² 1I_{Yi,n∈]x0,xN]}) +g∞E(²_i,n1I_{i,n∈]x0,xN]}) +

N−1 j=0

E(f_j(Y_i,n) [2g∞∧ g∞|Yi,n−x_j|])

+

N−1 j=0

E(f_j(_i,n) [2g_∞∧ g_∞|_i,n−x_j|]). (3.2)

Define the random functions h_i−1,i+1(x) = h(Y_1,n+. . . +Y_i−1,n+x+_i+1,n +. . .+_Kn,n) and g_i(x) = ₁

0(1−t)h_i−1,i+1(tx)dt. From Lindeberg’s decomposition and the independence of (Y_i,n) and (_i,n), we obtain that

E

h _K

n

i=1

Y_i,n

−E

h _K

n

i=1

_i,n

=

Kn

i=1

E

Y_i,n² g_i(Y_i,n)

−^Kn

i=1

E(²_i,ng_i(_i,n)). (3.3)

Clearlyg_i is bounded by 1/2h∞and is 1/6h⁽³⁾∞-lipschitz. Lemma 1 follows from (3.2) and (3.3).