Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays

In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n, n \geq 1}$ of random variables with values in $\bR \verb2\2 \{0\}$ (which satisfy some asymptotic dependence conditions), and the corresponding sequence $(N_{n})_{n\geq 1}$ of point processes, where $N_{n}$ has the points $X_{j,n}, 1\leq j \leq n$. Our main result identifies some explicit conditions for the convergence of the sequence $(N_{n})_{n \geq 1}$, in terms of the probabilistic behavior of the variables in the array.


Introduction
The study of the asymptotic behavior of the sum (or the maximum) of the row variables in an array (X j,n ) 1≤j≤n,n≥1 is one of oldest problem in probability theory. When the variables are independent on each row, classical results identify the limit to have an infinitely divisible distribution in the case of the sum (see [9]), and a max-infinitely divisible distribution, in the case of the maximum (see [3]). A crucial observation, which can be traced back to [17], [22] (in the case of the maximum), and [20] (in the case of the sum) is that these results are deeply connected to the convergence in distribution of the sequence N n = n j=1 δ Xj,n , n ≥ 1 of point processes to a Poisson process N . (See Section 5.3 of [18] and Section 7.2 of [19], for a modern account on this subject.) Subsequent investigations showed that a similar connection exists in the case of arrays which possess a row-wise dependence structure (e.g. [8]). The most interesting case arises when X i,n = X i /a n , where (X i ) i≥1 is a (dependent) stationary sequence with regularly varying tails and (a n ) n is a sequence of real numbers such that nP (|X 1 | > a n ) → 1 (see [7] and the references therein). In the dependent case, the limit N may not be a Poisson process, but belongs to the class of infinitely divisible point processes (under generally weak assumptions). These findings reveal that the separate study of the point process convergence is an important topic, which may yield new asymptotic results for triangular arrays.
The rows of the array also possess an "anti-clustering" property (AC), which specifies the dependence structure within a small block. Intuitively, under (AC), it becomes improbable to find two points X j,n , X k,n whose indices j, k are situated in the same small block at a distance larger than a fixed value m, and whose values (in modulus) exceed a fixed threshold ε > 0. Condition (AC) appeared, in various forms, in the literature related to the asymptotic behavior of the maximum (e.g. [14], [15]) or the sum (e.g. [5], [6], [4]). In addition, we assume the usual asymptotic negligibility (AN) condition for X 1,n .
Our main result says that under (AD-1), (AC) and (AN), the convergence N n d → N , where N is an infinitely divisible point process, reduces to the convergence of: where A m,n is the event that at least k i among X 1,n , . . . , X m,n lie in B i , for all i = 1, . . . , d (for arbitrary d, k 1 , . . . , k d ∈ N and compact sets B 1 , . . . , B d ).
The novelty of this result compared to the existing results (e.g. Theorem 2.6 of [1]), is the fact that the quantities appearing in (1) and (2) speak explicitly about the probabilistic behavior of the variables in the array.
The article is organized as follows. In Section 2, we give the statements of the main result (Theorem 2.5) and a preliminary result (Theorem 2.4). Section 3 is dedicated to the proof of these two results. Section 4 contains a separate result about the extremal index of a stationary sequence, whose proof is related to some of the methods presented in this article.
If E is a locally compact Hausdorff space with a countable basis (LCCB), we let B be the class of all relatively compact Borel sets in E, and C + K (E) be the class of continuous functions f : E → R + with compact support. We let M p (E) be the class of Radon measures on E with values in Z + (endowed with the topology of vague convergence), and M p (E) be the associated Borel σ-field. For µ ∈ M p (E) and f ∈ C + K (E), we denote µ(f ) = E f (x)µ(dx). We denote by o the null measure.
Let (Ω, K, P ) be a probability space. A measurable map N : By Theorem 6.1 of [13], the Laplace functional of an infinitely divisible point process is given by: where λ is a measure on M p (E)\{o}, called the canonical measure of N . All the point processes considered in this article have their points in R\{0}. For technical reasons, we embed R\{0} into the space E = [−∞, ∞]\{0}. Let B be the class of relatively compact sets in E. Note that We consider a triangular array (X j,n ) j≤n,n≥1 of random variables with values in R\{0}, such that (X j,n ) j≤n is a strictly stationary sequence, for any n ≥ 1. (ii) condition (AD-1) if there exists (r n ) n ⊂ N with r n → ∞ and k n = [n/r n ] → ∞, such that: In this case, N is an infinitely divisible point process with canonical measure λ.
is satisfied by arrays whose row-wise dependence structure is of mixing type (see e.g. (iii) When X j,n = X j /u n and m 0 = 1, condition (AC) is known in the literature as Leadbetter's condition D ′ ({u n }) (see [14]).
(iii) A condition similar to (AC) was used in [4], [5] and [6] for obtaining the convergence of the partial sum sequence to an infinitely divisible random variable (with finite variance).
As in [7], Recall that x is a fixed atom of a point process N if P (N {x} > 0) > 0. To simplify the writing, we introduce some additional notation. If x > 0 and λ is a measure on and J x,λ be the class of sets The following result is a refinement of Theorem 3.6 of [2].
Theorem 2.4 Suppose that (X j,n ) 1≤j≤n,n≥1 satisfies (AN) and (AD-1) (with sequences (r n ) n and (k n ) n ). Let N be an infinitely divisible point process on R\{0} with canonical measure λ. Let D be the set of fixed atoms of N and The following statements are equivalent: We have λ(M c 0 ) = 0, and the following two conditions hold: and for any set M ∈ J x,λ .
For each 1 ≤ m ≤ n, let N m,n = m j=1 δ Xj,n and M m,n = max j≤m |X j,n |, with the convention that M 0,n = 0. The next theorem is the main result of this article, and gives an explicit form for conditions (a) and (b), under the additional anti-clustering condition (AC).
Theorem 2.5 Let (X j,n ) 1≤j≤n,n≥1 and N be as in Theorem 2.4. Suppose in addition that (AC) holds, with the same sequence (r n ) n as in (AD-1).
The following statements are equivalent: We have λ(M c 0 ) = 0 and the following two conditions hold: Remark 2.7 Suppose that m 0 = 1 in Theorem 2.5. One can prove that in this case, the limit N is a Poisson process of intensity ν given by: 3 The Proofs

Proof of Theorem 2.4
Before giving the proof, we need some preliminary results.
Then for any B 1 , . . . , B d ∈ B N and for any d ≥ 1. Since these random vectors have values in Z d + , the previous convergence in distribution is equivalent to: for any k 1 , . . . , k d ∈ Z + , which is in turn equivalent to for any k 1 , . . . , k d ∈ Z + . Finally, it suffices to consider only integers k i ≥ 1 since, if there exists a set I ⊂ {1, . . . , d} such that k i = 0 for all i ∈ I and k i ≥ 1 for i ∈ I, then P (N n ( Proof of Theorem 2.4: Note that {max j≤rn |X j,n | > x} = {N rn,n ∈ M x }. Suppose that (i) holds. As in the proof of Theorem 3.6 of [2], it follows that λ(M c 0 ) = 0 and (a) holds. Moreover, we have P n,x w → P x where P n,x and P x are probability measures on M p (E) defined by: Therefore, P n,x (M ) → P x (M ) for any M ∈ M p (E) with P x (∂M ) = 0. Since k n P (N rn,n ∈ M x ) → λ(M x ) (by (a)), it follows that for any M ∈ M p (E) with λ(∂M ∩ M x ) = 0.

In particular, (3) holds for a set
To see this, note that by Lemma 3.1, Suppose that (ii) holds. As in the proof of Theorem 3.6 of [2], it suffices to show that P n,x w → P x . This follows by Lemma 3.2, since the class of sets B ∈ B which satisfy: P x ({µ ∈ M p (E); µ(∂B) > 0}) = 0 coincides with B x,λ .

Proof of Theorem 2.5
We begin with an auxiliary result, which is of independent interest.

Lemma 3.3
Let h : R d → R be a twice continuously differentiable function, such that Let (Y i ) i≥1 be a strictly stationary sequence of d-dimensional random vectors Proof: As in Lemma 3.2 of [12] (see also Theorem 2.6 of [1]), we have: where the second equality is due to the stationarity of (Y i ) i . Taking the difference, we get: Clearly For treating I 2 , we use the Taylor's formula (with integral remainder) for twice continuously differentiable functions f : R d → R: We get: Taking the difference of the last two equations, and using (5) for f = ∂h/∂x i with i = 1, . . . , d, we obtain: From here we conclude that: which yields the desired estimate for I 2 .
Proposition 3.4 Let E be a LCCB space. For each n ≥ 1, let (X j,n ) j≤n be a strictly stationary sequence of E-valued random variables, such that: Suppose that there exists (r n ) n ⊂ N with r n → ∞ and k n = [n/r n ] → ∞, such that: where m 0 =: {m ∈ Z + ; lim n→∞ n rn j=m+1 P (X 1,n ∈ B, X j,n ∈ B) = 0, for all B ∈ B}. Let N m,n = Note that: For any n ≥ 1, we consider strictly stationary sequence of d-dimensional random vectors {Y j,n = (Y Using Lemma 3.3, and letting C = D 2 h ∞ , we obtain: k+m,n ) =: I (1) m,n + CI (2) m,n Using the fact that h(x) ≤ d i=1 x i , and the stationary of (X j,n ) j≤n , nP (X 1,n ∈ B i ).
From (6), it follows that lim n→∞ I Using the stationarity of (X j,n ) j≤n , and letting P (X 1,n ∈ B, X j,n ∈ B).  for any x > 0, and for any set

The extremal index
In this section, we give a recipe for calculating the extremal index of a stationary sequence, using a method which is similar to that used for proving Theorem 2.5, in a simplified context. Although this recipe (given by Theorem 4.5 below) seems to be known in the literature (see [15], [16], [21]), we decided to include it here, since we could not find a direct reference for its proof.
We recall the following definition.
Definition 4.1 Let (X j ) j≥1 be a strictly stationary sequence of random variables. The extremal index of the sequence (X j ) j≥1 , if it exists, is a real number θ with the following property: for any τ > 0, there exists a sequence (u In particular, for τ = 1, we denote u (1) n = u n , and we have nP (X 1 > u n ) → 1 and P (max It is clear that if it exists, θ ∈ [0, 1].

Remark 4.2
The extremal index of an i.i.d sequence exists and is equal to 1.
The following definition was originally considered in [15].

Definition 4.3
We say that (X j ) j≥1 satisfies condition (AIM) (or admits an asymptotic independence representation for the maximum) if there exists (r n ) n ⊂ N with r n → ∞ and k n = [n/r n ] → ∞, such that: where the supremum ranges over all disjoint subsets I, J of {1, . . . , n}, which are separated by a block of length greater of equal than m.
The following theorem is the main result of this section.
Theorem 4.5 Let (X j ) j≥1 be a strictly stationary sequence whose extremal index θ exists, and (u n ) n be a sequence of real numbers satisfying (12). Suppose that (X j ) j≥1 satisfies (AIM), and in addition, where m 0 := inf{m ∈ Z + ; lim n→∞ n rn j=m+1 P (X 1 > u n , X j > u n ) = 0}.
Due to the stationarity, and the fact that nP (X 1 > u n ) → 1, (14) can be written as: Remark 4.6 Let (Y i ) i≥1 be a sequence of i.i.d. random variables and X i = max(Y i , · · · , Y i+m−1 ). Then (X i ) i≥1 satisfies condition (13), since it is an mdependent sequence. A direct calculation shows that the extremal index of (X i ) i≥1 exists and is equal to 1/m, which can be deduced also from (14).
The proof of Theorem 4.5 is based on some intermediate results.
Proposition 4.7 Let (X j ) j≥1 be a strictly stationary sequence whose extremal index θ exists, and (u n ) n be a sequence of real numbers satisfying (12). If (X j ) j≥1 satisfies (AIM), then k n P (max j≤rn X j > u n ) → θ.
Proof: Due to (AIM), P (max j≤rn X j ≤ u n ) kn → e −θ . The result follows, since P (max j≤rn X j ≤ u n ) kn = 1 − k n P (max j≤rn X j > u n ) k n kn . Proposition 4.8 Let (X j ) j≥1 be a strictly stationary sequence such that: lim sup n→∞ nP (X 1 > u n ) < ∞.
Suppose that there exists (r n ) n ⊂ N with r n → ∞ and k n = [n/r n ] → ∞, such that (13)  We omit the details.
Proof of Theorem 4.5: The result follows from Proposition 4.7 and Proposition 4.8, using the fact that: