in PROBABILITY
EXPLICIT CONDITIONS FOR THE CONVERGENCE OF POINT
PRO-CESSES ASSOCIATED TO STATIONARY ARRAYS
RALUCA BALAN1
Corresponding author. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada.
email: rbalan@uottawa.ca
SANA LOUHICHI2
Laboratoire de mathématiques, Équipe de Probabilités, Statistique et modélisation, Université de Paris-Sud, Bât. 425, F-91405 Orsay Cedex, France.
email: Sana.Louhichi@math.u-psud.fr
SubmittedDecember 7, 2009, accepted in final formAugust 6, 2010 AMS 2000 Subject classification: Primary 60E07, 60G55; secondary 60G10, 60G57
Keywords: infinite divisibility, point process, asymptotic independence, weak convergence, ex-tremal index
Abstract
In this article, we consider a stationary array (Xj,n)1≤j≤n,n≥1of random variables (which satisfy
some asymptotic independence conditions), and the corresponding sequence (Nn)n≥1 of point
processes, where Nn has the points Xj,n, 1 ≤ j ≤ n. Our main result identifies some explicit conditions for the convergence of the sequence(Nn)n≥1, in terms of the probabilistic behavior of the variables in the array.
1
Introduction
The study of the asymptotic behavior of the sum (or the maximum) of the row variables in an array(Xj,n)1≤j≤n,n≥1 is one of the oldest problem in probability theory. When the variables are
independent on each row, classical results identify the limit to have an infinitely divisible distribu-tion in the case of the sum (see[10]), and a max-infinitely divisible distribution, in the case of the maximum (see[3]). A crucial observation, which can be traced back to[20],[27](in the case of the maximum), and[23](in the case of the sum) is that these results are deeply connected to the convergence in distribution of the sequenceNn=
Pn
j=1δXj,n,n≥1 of point processes to a Poisson
processN. (See Section 5.3 of[21]and Section 7.2 of[22], 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.[9]). The most interesting case arises whenXi,n=
Xi/an, where(Xi)i≥1is a (dependent) stationary sequence with regularly varying tails and(an)n
1SUPPORTED BY A GRANT FROM THE NATURAL SCIENCES AND ENGINEERING RESEARCH COUNCIL OF CANADA. 2PARTIALLY SUPPORTED BY THE GRANT ANR-08-BLAN-0314-01.
is a sequence of real numbers such thatnP(|X1|>an)→1 (see [8]and the references therein). The behavior of the maxima of m-dependent sequences exhibiting clustering of big values was first studied in [17], whereas the first formula for the limits of sums of m-dependent heavy-tailed random variables was obtained in[14]. 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.
In the present article, we consider an array(Xj,n)1≤j≤n,n≥1whose row variables are asymptotically independent, in the sense that the block (X1,n, . . . ,Xn,n) behaves asymptotically askn“smaller” i.i.d. blocks, a small block having the same distribution as(X1,n, . . . ,Xrn,n), withn∼ rnkn. This condition, that we call here (AD-1), was considered by many authors (e.g. [12] [13],[8],[11], [1]).
The rows of the array also possess an “anti-clustering” property (AC), which specifies the depen-dence structure within a small block. Intuitively, under (AC), it becomes improbable to find two points Xj,n,Xk,nwhose indices j,kare situated in the same small block at a distance larger than a fixed valuem, and whose values (in modulus) exceed a fixed thresholdǫ >0. Condition (AC) appeared, in various forms, in the literature related to the asymptotic behavior of the maximum (e.g. [16], [18]) or the sum (e.g. [6], [7], [4]). In addition, we assume the usual asymptotic negligibility (AN) condition forX1,n.
Our main result says that under (AD-1), (AC) and (AN), the convergenceNn d
→N, whereNis an infinitely divisible point process, reduces to the convergence of:
nP( max
1≤j≤m−1|Xj,n| ≤x,Xm,n>x), and (1) n[P(Am,n, max
1≤j≤m|Xj,n|>x)−P(Am−1,n, max1≤j≤m−1|Xj,n|>x)], (2)
where Am,n is the event that at least li among X1,n, . . . ,Xm,n lie in Bi, for all i = 1, . . . ,d (for arbitraryd,l1, . . . ,ld ∈Nand compact setsB
1, . . . ,Bd).
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) speakexplicitlyabout 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 (Theo-rem 2.5) and a preliminary result (Theo(Theo-rem 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.
2
The Main Results
We begin by introducing the terminology and the notation. Our main reference is[15]. We denote
R
+= [0,∞),Z+={0, 1, 2, . . .}andN={1, 2, . . .}.
IfEis a locally compact Hausdorff space with a countable basis (LCCB), we letB be the class of all relatively compact Borel sets inE, andCK+(E)be the class of continuous functions f :E→R+
with compact support. We let Mp(E)be the class of Radon measures on E with values in Z+
(endowed with the topology of vague convergence), andMp(E)be the associated Borelσ-field. Forµ∈Mp(E)andf ∈C+
K(E), we denoteµ(f) =
R
A point process N isinfinitely divisibleif for any k ≥ 1, there exist some i.i.d. point processes
(Ni,k)1≤i≤ksuch thatN=d Pk
i=1Ni,k. By Theorem 6.1 of[15], the Laplace functional of an infinitely divisible point process is given by:
LN(f) =exp
All the point processes considered in this article have their points inR\{0}. For technical reasons,
we embedR\{0}into the spaceE= [−∞,∞]\{0}. LetBbe the class of relatively compact sets
does, and the limits are the same.
(ii) Condition (AN) is an asymptotic negligibility condition which ensures that(N˜i,n)i≤kn,n≥1is a
In this case,Nis an infinitely divisible point process with canonical measureλ.
Remark 2.3. (i) Condition (AD-1) is satisfied by arrays whose row-wise dependence structure is of mixing type (see e.g. Lemma 5.1 of[1]).
(iii) When Xj,n= Xj/un andm0 =1, condition (AC) is known in the literature as Leadbetter’s
conditionD′({un})(see[16]).
(iv) Condition (AC) coincides with (22) of[1], and originates in the work of[6]and[7]. Note that, by the stationarity of the array,
P(
kn [
i=1
[
j,l∈Bi,n,j−l≥m
{|Xl,n|> ǫ,|Xj,n|> ǫ})≤knrn rn X
j=m+1
P(|X1,n|> ǫ,|Xj,n|> ǫ),
where Bi,n = {(i−1)rn+1, . . . ,i rn} for i = 1, . . . ,kn. Therefore, (AC) is an asymptotic anti-clustering condition which implies that, whenmis close tom0andnis large enough, it is unlikely that there will be two indicesj>lin the same blockBi,n, located at a minimum distancemof each other, and whose corresponding measurements Xj,nandXl,nexceed (in modulus) the threshold
ǫ. Condition (2.10) of[4]is similar to (AC) and was used for obtaining the convergence of the partial sum sequence to an infinitely divisible random variable (with infinite variance). We refer to[4]for a detailed discussion of this condition.
As in [8], letM0 = {µ ∈Mp(R\{0});µ =6 o, ∃ x ∈(0,∞)such that µ([−x,x]c) = 0}. If µ=
P
j≥1δtj∈M0, we letxµ:=supj≥1|tj|<∞. For eachx>0, let
Mx={µ∈M0;µ([−x,x]c)>0}={µ∈M0;xµ>x}.
Recall thatx is afixed atomof a point processN ifP(N{x}>0)>0. To simplify the writing, we introduce some additional notation. Ifx>0 andλis a measure onMp(E)withλ(M0c) =0, we let
Bx,λ={B∈ B;λ({µ∈Mx;µ(∂B)>0}) =0},
andJx,λbe the class of setsM=∩di=1{µ∈Mp(E);µ(Bi)≥li}for someBi∈ Bx,λ,li≥1 (integers) andd≥1.
The following result is a refinement of Theorem 3.6 of[2].
Theorem 2.4. Suppose that(Xj,n)1≤j≤n,n≥1satisfies (AN) and (AD-1) (with sequences(rn)and(kn)).
Let N be an infinitely divisible point process onR\{0}with canonical measureλ. Let D be the set of
fixed atoms of N and D′={x>0;x∈D or −x∈D}. The following statements are equivalent:
(i) Nn d →N ;
(ii) We haveλ(M0c) =0, and the following two conditions hold:
(a) knP(max
j≤rn
|Xj,n|>x)→λ(Mx), for any x>0,x6∈D′,
(b) knP(Nrn,n∈M, max j≤rn
|Xj,n|>x)→λ(M∩Mx), for any x>0,x6∈D′
and for any set M∈ Jx,λ. For each 1≤ m≤n, let Nm,n=
Pm
j=1δXj,n and Mm,n =maxj≤m|Xj,n|, with the convention that M0,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(Xj,n)1≤j≤n,n≥1and N be as in Theorem 2.4. Suppose in addition that (AC) holds, with the same sequence(rn)nas in (AD-1).
(i) Nn→d N ; (ii) We haveλ(Mc
0) =0and the following two conditions hold:
(a′) lim
m→m0
lim sup n→∞ |
n[P(Mm,n>x)−P(Mm−1,n>x)]−λ(Mx)|=0, for any x>0,x6∈D′,
(b′) lim
m→m0
lim sup n→∞ |
n[P(Nm,n∈M,Mm,n>x)−P(Nm−1,n∈M,Mm−1,n>x)] −λ(M∩Mx)|=0, for any x>0,x6∈D′and for any set M ∈ Jx,λ.
Remark 2.6. Note that
P(Mm,n>x)−P(Mm−1,n>x) =P( max
1≤j≤m−1|Xj,n| ≤x,|Xm,n|>x).
Remark 2.7. Suppose thatm0=1 in Theorem 2.5. One can prove that in this case, the limitNis
a Poisson process of intensityνgiven by:
ν(B) =λ({µ∈Mp(E)\{o};µ(B) =1}), ∀B∈ B.
3
The Proofs
3.1
Proof of Theorem 2.4
Before giving the proof, we need some preliminary results.
Lemma 3.1. Let E be a LCCB space and M=∩d
i=1{µ∈Mp(E);µ(Bi)≥li}for some Bi∈ B, li≥1
(integers) and d≥1. Then:
(i) M is closed (with respect to the vague topology); (ii)∂M⊂ ∪d
i=1{µ∈Mp(E);µ(∂Bi)>0}.
Proof: Note that∂M⊂ ∪d
i=1∂Mi, where Mi ={µ∈Mp(E);µ(Bi)≥li}. Since the finite intersec-tion of closed sets is a closed set, it suffices to consider the cased=1, i.e. M={µ∈Mp(E);µ(B)≥
l}for someB∈ Bandl≥1.
(i) Let(µn)n⊂M be such thatµn→v µ. Ifµ(∂B) =0, thenµn(B)→µ(B), and sinceµn(B)≥lfor alln, it follows thatµ(B)≥l. If not, we proceed as in the proof of Lemma 3.15 of[21]. LetBδbe a
δ-swelling ofB. ThenS={δ∈(0,δ0];µ(∂Bδ)>0}is a countable set. By the previous argument,
µ(Bδ)≥lfor allδ∈(0,δ
0]\S. Let(δn)n∈(0,δ0]\Sbe such thatδn↓0. Sinceµ(Bδn)≥lfor all
n, andµ(Bδn)↓µ(B), it follows thatµ(B)≥l, i.e.µ∈M.
(ii) By part (i), ∂M =M¯\Mo = M∩(Mo)c. We will prove that∂M ⊂ {µ∈M;µ(∂B)>0}, or equivalently
A:={µ∈M;µ(∂B) =0} ⊂Mo.
SinceMois the largest open set included inM andA⊂M, it suffices to show thatAis open. Let
µ ∈Aand(µn)n ⊂ Mp(E)be such thatµn v
→ µ. Thenµn(B)→ µ(B), and sinceµ(B)≥ l and {µn(B)}nare integers, it follows thatµn(B)≥lfor alln≥n1, for somen1.
Lemma 3.2. Let E be a LCCB space and(Qn)n,Q be probability measures on Mp(E). LetBQbe the
class of all sets B∈ Bwhich satisfy:
Q({µ∈Mp(E);µ(∂B)>0}) =0,
andJQbe the class of sets M=∩d
i=1{µ∈Mp(E);µ(Bi)≥li}for some Bi∈ BQ, li≥1(integers) and
d≥1. Then Qn
w
→Q if and only if Qn(M)→Q(M)for all M∈ JQ.
Proof: Let (Nn)n,N be point processes onE, defined on a probability space(Ω,F,P), such that
P◦N−1
n =Qnfor alln, andP◦N−1=Q. Note thatBQ=BN:={B∈ B;N(∂B) =0 a.s.}. By definition,Nn→d Nif and only ifQn→w Q. By Theorem 4.2 of[15],Nn→d N if and only if
(Nn(B1), . . . ,Nn(Bd)) d
→(N(B1), . . . ,N(Bd))
for any B1, . . . ,Bd ∈ BN and for anyd ≥1. Since these random vectors have values inZd+, the
previous convergence in distribution is equivalent to:
P(Nn(B1) =l1, . . . ,Nn(Bd) =ld)→P(N(B1) =l1, . . . ,N(Bd) =ld) for anyl1, . . . ,ld∈Z+, which is in turn equivalent to
P(Nn(B1)≥l1, . . . ,Nn(Bd)≥ld)→P(N(B1)≥l1, . . . ,N(Bd)≥ld)
for anyl1, . . . ,ld∈Z+. Finally, it suffices to consider only integersli≥1 since, if there exists a set
I ⊂ {1, . . . ,d}such thatli=0 for alli∈I andli≥1 fori6∈I, thenP(Nn(B1)≥l1, . . . ,Nn(Bd)≥
ld) =P(Nn(Bi)≥li,i6∈I)→P(N(Bi)≥li,i6∈I) =P(N(B1)≥l1, . . . ,N(Bd)≥ld).
Proof of Theorem 2.4:Note that{maxj≤rn|Xj,n|>x}={Nrn,n∈Mx}.
Suppose that (i) holds. As in the proof of Theorem 3.6 of [2], it follows thatλ(Mc
0) =0 and (a)
holds. Moreover, we havePn,x w
→PxwherePn,xandPxare probability measures onMp(E)defined by:
Pn,x(M) =
knP(Nrn,n∈M∩Mx)
knP(Nrn,n∈Mx) and Px(M) =
λ(M∩Mx) λ(Mx) .
Therefore, Pn,x(M)→ Px(M)for any M ∈ Mp(E)with Px(∂M) =0. SinceknP(Nrn,n∈Mx)→ λ(Mx)(by (a)), it follows that
knP(Nrn,n∈M∩Mx)→λ(M∩Mx), (3) for anyM∈ Mp(E)withλ(∂M∩Mx) =0.
In particular, (3) holds for a setM=∩di=1{µ∈Mp(E);µ(Bi)≥li}, withBi∈ Bx,λ,li≥1 (integers) andd≥1. To see this, note that by Lemma 3.1,∂M∩Mx⊂ ∪di=1{µ∈Mx;µ(∂Bi)>0}, and hence
λ(∂M∩Mx)≤ d
X
i=1
λ({µ∈Mx;µ(∂Bi)>0}) =0.
Suppose that (ii) holds. As in the proof of Theorem 3.6 of[2], it suffices to show thatPn,x w →Px. This follows by Lemma 3.2, since the class of setsB∈ Bwhich satisfy:
3.2
Proof of Theorem 2.5
We begin with an auxiliary result, which is of independent interest.
Lemma 3.3. Let h:Rd→Rbe a twice continuously differentiable function, such that
kD2hk∞:= max
Let(Yi)i≥1be a strictly stationary sequence of d-dimensional random vectors withYi= (Y
(1)
where the second equality is due to the stationarity of(Yi)i. Taking the difference, we get:
E[h(Sr)]−r E[h(Sm)−h(Sm−1)] =mE[h(Sm−1)]−(m−1)E[h(Sm)]+ remainder) for twice continuously differentiable functions f :Rd →R:
Taking the difference of the last two equations, and using (5) for f =∂h/∂xi withi=1, . . . ,d,
which yields the desired estimate forI2.
Proposition 3.4. Let E be a LCCB space. For each n≥1, let(Xj,n)j≤nbe a strictly stationary sequence
of E-valued random variables, such that:
lim sup
+→R+ be a twice continuously differentiable function which satisfies (4), such
thath(x1, . . . ,xd)≤x1+. . .+xdfor all(x1, . . . ,xd)∈R+d, and
For anyn≥1, we consider strictly stationary sequence ofd-dimensional random vectors{Yj,n= (Yj(,1n), . . . ,Yj(,nd)), 1≤ j≤n}defined by:
From, (9), (10) and (11), it follows that:
lim m→m0
lim sup n→∞
kn|P(Nrn,n∈M)−rn[P(Nm,n∈M)−P(Nm−1,n∈M)]|=0. Note that limn→∞(n−knrn)|P(Nm,n∈M)−P(Nm−1,n∈M)|=0 for allm, and hence
lim m→m0
lim sup n→∞
(n−knrn)|P(Nm,n∈M)−P(Nm−1,n∈M)|=0.
The conclusion follows.
Corollary 3.5. For each n≥1, let(Xj,n)1≤j≤nbe a strictly stationary sequence of random variables
with values inR\{0}. Suppose that(Xj,n)1≤j≤n,n≥1satisfies (AN) and (AC).
For any1≤m≤n, let Nm,n=
Pm
j=1δXj,nand Mm,n=maxj≤m|Xj,n|. Then,
lim m→m0
lim sup n→∞ |
knP(Mrn,n>x)−n[P(Mm,n>x)−P(Mm−1,n>x)]|=0
lim m→m0
lim sup n→∞ |
knP(Nrn,n∈M,Mrn,n>x)−n[P(Nm,n∈M,Mm,n>x)−
P(Nm−1,n∈M,Mm−1,n>x)]|=0,
for any x>0, and for any set M=∩d
i=1{µ∈Mp(E);µ(Bi)≥li}, with Bi∈ B, li≥1(integers) and
d≥1.
Proof: Since{Mm,n > x} = {Nm,n([−x,x]c)≥ 1} for any 1≤ m ≤ n, the result follows from Proposition 3.4.
Proof of Theorem 2.5:The result follows from Theorem 2.4 and Corollary 3.5.
4
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 [18], [19], [26]), 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(Xj)j≥1be a strictly stationary sequence of random variables. The extremal indexof the sequence (Xj)j≥1, if it exists, is a real numberθ with the following property: for anyτ >0, there exists a sequence(u(τ)
n )n⊂Rsuch thatnP(X1>u(nτ))→τandP(maxj≤nXj≤
u(nτ))→e−τθ.
In particular, forτ=1, we denoteu(1)
n =un, and we have
nP(X1>un)→1 and P(max
j≤n Xj≤un)→e
−θ. (12)
It is clear that if it exists,θ∈[0, 1].
Definition 4.3. We say that(Xj)j≥1satisfiescondition (AIM)(or admits an asymptotic indepen-dence representation for the maximum) if there exists(rn)n⊂Nwithrn→ ∞andkn= [n/rn]→ ∞, such that:
P(max
j≤n Xj≤un)−P(maxj≤rn
Xj≤un)kn
→0.
Remark 4.4. It is known that (Leadbetter’s) conditionD({un})implies (AIM) (see Lemma 2.1 of [16]). Recall that(ξj)jsatisfies conditionD({un})if there exists a sequence(mn)n⊂N, such that
mn=o(n)andαn(mn)→0, where
αn(m) =sup I,J |
P(max
j∈I Xj≤un, maxj∈J Xj≤un)−P(maxj∈I Xj≤un)P(maxj∈J Xj≤un)|,
where the supremum ranges over all disjoint subsets I,J of{1, . . . ,n}, which are separated by a block of length greater of equal thanm.
The following theorem is the main result of this section.
Theorem 4.5. Let(Xj)j≥1be a strictly stationary sequence whose extremal indexθexists, and(un)n
be a sequence of real numbers satisfying (12). Suppose that(Xj)j≥1satisfies (AIM), and in addition,
lim m→m0
lim sup n→∞ n
rn X
j=m+1
P(X1>un,Xj>un) =0, (13)
where m0:=inf{m∈Z+; limn→∞n
Prn
j=m+1P(X1>un,Xj>un) =0}.
Then
θ= lim m→m0
lim sup n→∞
nP( max
1≤j≤m−1Xj≤un,Xm>un). (14)
Due to the stationarity, and the fact thatnP(X1>un)→1, (14) can be written as:
θ = lim m→m0
lim sup
n→∞ nP
(max
2≤j≤mXj≤un,X1>un)
= lim
m→m0
lim sup n→∞
P(max
2≤j≤mXj≤un|X1>un),
which coincides with (2.3) of[26]. We refer to formula (5) in[24]and to Theorem 3.1 in[25] for a similar expression ofθ. We refer also to[5], Chapter 10, for an overview on the extremal index.
Remark 4.6. Let (Yi)i≥1be a sequence of i.i.d. random variables andXi=max(Yi,· · ·,Yi+m−1).
Then (Xi)i≥1 satisfies condition (13), since it is anm-dependent sequence. A direct calculation
shows that the extremal index of(Xi)i≥1exists 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.
Proof:Due to (AIM),P(maxj≤rnXj≤un)
kn→e−θ. The result follows, since
P(max j≤rn
Xj≤un)kn=
1−knP(maxj≤rnXj>un) kn
kn
.
Proposition 4.8. Let(Xj)j≥1be a strictly stationary sequence such that:
lim sup n→∞
nP(X1>un)<∞.
Suppose that there exists(rn)n⊂Nwith rn→ ∞and kn= [n/rn]→ ∞, such that (13) holds. Then
lim m→m0
lim sup n→∞ |
knP(max j≤rn
Xj>un)−n[P(max
j≤mXj>un)−P(j≤maxm−1Xj>un)]|=0.
Proof: The argument is the same as in the proof of Proposition 3.4, using Lemma 3.3. More precisely, we let h : R+ → R+ be a twice continuously differentiable such that kh′′k∞ < ∞,
h(0) =0,h(1) =1 ify≥1, andh(x)≤xfor allx∈R+. Thenh(x) =1{x≥1} for allx∈Z+, and
P(max
j≤m Xj>un) = E[1{ Pm
j=11{X j>un}≥1}] =E[h(
m
X
j=1
1{Xj>un})].
We omit the details.
Proof of Theorem 4.5:The result follows from Proposition 4.7 and Proposition 4.8, using the fact that:
P(max
j≤m−1Xj≤un,Xm>un) =P(maxj≤mXj>un)−P(jmax≤m−1Xj>un).
Acknowledgement.The first author would like to thank the second author and Laboratoire de mathématiques de l’Université de Paris-sud (Équipe de Probabilités, Statistique et modélisation) for their warm hospitality. The authors would like to thank J. Segers for drawing their attention to references[5],[24]and[25].
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