getdoc40ab. 189KB Jun 04 2011 12:04:18 AM

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 (X_j,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(N_n)_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

1_{SUPPORTED BY A GRANT FROM THE NATURAL SCIENCES AND ENGINEERING RESEARCH COUNCIL OF CANADA.} 2_{PARTIALLY 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(X_j,n)_{1≤j≤n,n≥1}whose row variables are asymptotically independent, in the sense that the block (X_1,n, . . . ,X_n,n) behaves asymptotically ask_n“smaller” i.i.d. blocks, a small block having the same distribution as(X_1,n, . . . ,X_rn,n), withn∼ r_nk_n. 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(A_m,n, max

1≤j≤m|Xj,n|>x)−P(Am−1,n, max1≤j≤m−1|Xj,n|>x)], (2)

where A_m,n is the event that at least l_i among X_1,n, . . . ,X_m,n lie in B_i, for all i = 1, . . . ,d (for arbitraryd,l₁, . . . ,l_d ∈N_{and 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, andC_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), andMp(E)be the associated Borelσ-field. Forµ∈M_p(E)andf ∈C+

K(E), we denoteµ(f) =

R

E f(x)µ(d x). We denote byothe null measure. Let(Ω,K,P)be a probability space. A measurable mapN:Ω→M_p(E)is called a point process. Its distributionP◦N−1is determined by the Laplace functionalLN(f) =E(e−N(f)),f ∈CK+(E).

A point process N isinfinitely divisibleif for any k ≥ 1, there exist some i.i.d. point processes

(N_i,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:

L_N(f) =exp

(

−

Z

Mp(E)\{o}

(1−e−µ(f))λ(dµ) )

, ∀f ∈C_K+(E), whereλis a measure onM_p(E)\{o}, called the canonical measure ofN.

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}

inE. Note that

[−x,x]c:= [−∞,−x)∪(x,∞]∈ B, for allx>0.

We consider a triangular array(X_j,n)_j≤n,n≥1of random variables with values inR_{\{0}, such that}

(X_j,n)_j≤nis a strictly stationary sequence, for anyn≥1.

Definition 2.1. The triangular array(X_j,n)₁≤j≤n,n≥1satisfies: (i)condition (AN)if

lim sup n→∞

nP(|X_1,n|> ǫ)<∞, for allǫ >0.

(ii)condition (AD-1)if there exists(r_n)_n⊂N_withr

n→ ∞andkn= [n/rn]→ ∞, such that: lim

n→∞

E

e−

Pn j=1f(Xj,n)

−nE

e−

P_rn

j=1f(Xj,n) okn

=0, for all f ∈C_K+(E).

(iii)condition (AC)if there exists(rn)n⊂Nwithrn→ ∞, such that: lim

m→m0

lim sup n→∞ n

rn X

j=m+1

P(|X1,n|> ǫ,|Xj,n|> ǫ) =0, for allǫ >0,

wherem0:=inf{m∈Z+; limn→∞n

Pr_n

j=m+1P(|X1,n|> ǫ,|Xj,n|> ǫ) =0, for allǫ >0}. We use the conventions: inf;=∞and lim_m→m0φ(m) =φ(m0)ifm0<∞.

Remark 2.2. (i) For eachn≥1, letNn=

Pn

j=1δXj,n and ˜Nn= Pkn

i=1N˜i,n, where(N˜i,n)i≤knare i.i.d.

copies of Nrn,n =

Prn

j=1δXj,n. Under (AD-1),(Nn)n converges in distribution if and only if (N˜n)n

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

null-arrayof point processes, i.e. P(N˜_1,n(B)>0)→ 0 for all B ∈ B. By Theorem 6.1 of[15] ˜

N_n→d N if and only if

knE(1−e−Nrn,n(f))→

Z

Mp(E)\{o}

(1−e−µ(f))λ(dµ), ∀f ∈C_K+(E). 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 B_i,n = {(i−1)r_n+1, . . . ,i r_n} for i = 1, . . . ,k_n. 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 blockB_i,n, located at a minimum distancemof each other, and whose corresponding measurements X_j,nandX_l,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

M_x={µ∈M₀;µ([−x,x]c)>0}={µ∈M₀;x_µ>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(X_j,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) k_nP(max

j≤rn

|X_j,n|>x)→λ(M_x), 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 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)_nas in (AD-1).

(i) N_n→d N ; (ii) We haveλ(Mc

0) =0and the following two conditions hold:

(a′) lim

m→m0

lim sup n→∞ |

n[P(M_m,n>x)−P(M_m−1,n>x)]−λ(M_x)|=0, for any x>0,x6∈D′,

(b′) lim

m→m0

lim sup n→∞ |

n[P(N_m,n∈M,M_m,n>x)−P(N_m−1,n∈M,Mm−1,n>x)] −λ(M∩M_x)|=0, for any x>0,x6∈D′and for any set M ∈ Jx,λ.

Remark 2.6. Note that

P(M_m,n>x)−P(M_m−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∈M_p(E);µ(B)≥

l}for someB∈ Bandl≥1.

(i) Let(µ_n)_n⊂M be such thatµ_n→v µ. Ifµ(∂B) =0, thenµ_n(B)→µ(B), and sinceµ_n(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(µ_n)_n ⊂ Mp(E)be such thatµn v

→ µ. Thenµ_n(B)→ µ(B), and sinceµ(B)≥ l and {µ_n(B)}nare integers, it follows thatµn(B)≥lfor alln≥n1, for somen1.

On the other hand, µ_n(∂B)→µ(∂B), since∂B ∈ B andµ(∂B) =0 (note that∂(∂B) =∂B). Sinceµ(∂B) =0 and{µ_n(∂B)}nare integers, it follows thatµn(∂B) =0 for alln≥n2, for some n2. Henceµn∈Afor alln≥max{n1,n2}.ƒ

Lemma 3.2. Let E be a LCCB space and(Q_n)_n,Q be probability measures on Mp(E). LetBQbe the

class of all sets B∈ Bwhich satisfy:

Q({µ∈M_p(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 (N_n)_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,N_n→d Nif and only ifQ_n→w Q. By Theorem 4.2 of[15],N_n→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(N_n(B₁)≥l₁, . . . ,N_n(B_d)≥l_d)→P(N(B₁)≥l₁, . . . ,N(B_d)≥l_d)

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).ƒ

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) =

k_nP(N_rn,n∈M∩M_x)

k_nP(N_rn,n∈M_x) and Px(M) =

λ(M∩M_x) λ(M_x) .

Therefore, Pn,x(M)→ Px(M)for any M ∈ Mp(E)with Px(∂M) =0. SinceknP(Nrn,n∈Mx)→ λ(M_x)(by (a)), it follows that

k_nP(N_rn,n∈M∩M_x)→λ(M∩M_x), (3) for anyM∈ Mp(E)withλ(∂M∩Mx) =0.

In particular, (3) holds for a setM=∩d_i=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∩M_x)≤ d

X

i=1

λ({µ∈M_x;µ(∂B_i)>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:

P_x({µ∈M_p(E);µ(∂B)>0}) =0 coincides withBx,λ.ƒ

3.2

Proof of Theorem 2.5

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

Lemma 3.3. Let h:Rd_→R_{be a twice continuously differentiable function, such that}

kD2hk∞:= max i,j=1,...,d_xsup_∈Rd

∂2h ∂xi∂xj

(x)

<∞. (4)

Let(Y_i)_i≥1be a strictly stationary sequence of d-dimensional random vectors withYi= (Y

(1)

i , . . . ,Y

(d)

i ).

LetSn=

Pn

i=1Yi for n≥1andS0=0. Then for any1≤m≤r,

|E[h(S_r)]−r E[h(S_m)−h(S_m−1)]| ≤m|E[h(S_m)] +E[h(S_m−1)]|+

kD2hk∞ rn−m

X

k=0

d

X

i,l=1

E|S(_ki)Y_k(₊l)_m|.

Proof:As in Lemma 3.2 of[13](see also Theorem 2.6 of[1]), we have:

E[h(S_r)] = E[h(S_m−1)] +

r−m

X

k=0

E[h(S_k+m)−h(S_k+m−1)] r E[h(Sm)−h(Sm−1)] = (m−1)E[h(Sm)−h(Sm−1)] +

r−m

X

k=0

E[h(S_k+m−Sk)−h(Sk+m−1−Sk)],

where the second equality is due to the stationarity of(Y_i)_i. Taking the difference, we get:

E[h(S_r)]−r E[h(S_m)−h(S_m−1)] =mE[h(S_m−1)]−(m−1)E[h(S_m)]+

r−m

X

k=0

E{[h(S_k+m)−h(Sk+m−Sk)]−[h(Sk+m−1)−h(Sk+m−1−Sk)]}=:I1+I2.

Clearly|I₁| ≤m|E[h(S_m−1)]+E[h(S_m)]|. For treatingI₂, we use the Taylor’s formula (with integral remainder) for twice continuously differentiable functions f :Rd _→R_:

f(x)−f(x₀) =

d

X

i=1

(x(i)−x₀(i)) Z1

0 ∂f ∂xi

(x−s(x−x₀))ds. (5)

We get:

h(S_k+m)−h(S_k+m−S_k) =

d

X

i=1 S(_ki)

Z1

0 ∂h ∂xi

(S_k+m−xS_k)d x,

h(Sk+m−1)−h(Sk+m−1−Sk) = d

X

i=1 S(_ki)

Z1

0 ∂h

Taking the difference of the last two equations, and using (5) for f =∂h/∂xi withi=1, . . . ,d, we obtain:

[h(S_k+m)−h(Sk+m−Sk)]−[h(Sk+m−1)−h(Sk+m−1−Sk)] = d

X

i=1 S(_ki)

Z1

0 _∂

h ∂xi

(S_k+m−xSk)−

∂h ∂xi

(S_k+m−1−xSk)

d x

=

d

X

i=1 S(_ki)

Z1

0

d

X

l=1 Y_k(₊l)_m

Z1

0 ∂2_h

∂x_i∂x_l((Sk+m−xSk)−θYk+m)dθd x.

From here we conclude that:

|[h(S_k+m)−h(S_k+m−S_k)]−[h(S_k+m−1)−h(S_k+m−1−S_k)]| ≤ kD2hk∞ d

X

i,l=1

|S_k(i)Y_k(₊l)_m|, which yields the desired estimate forI₂.ƒ

Proposition 3.4. Let E be a LCCB space. For each n≥1, let(X_j,n)_j≤nbe a strictly stationary sequence

of E-valued random variables, such that:

lim sup n→∞

nP(X_1,n∈B)<∞, for all B∈ B. (6)

Suppose that there exists(r_n)_n⊂N_{with r}

n→ ∞and kn= [n/rn]→ ∞, such that: lim

m→m0

lim sup n→∞

n

rn X

j=m+1

P(X_1,n∈B,X_j,n∈B) =0, for all B∈ B, (7)

where m0 =:{m∈Z+; limn→∞n

Prn

j=m+1P(X1,n∈B,Xj,n ∈B) =0, for all B∈ B}. Let Nm,n=

Pm

j=1δXj,n. Then

lim m→m0

lim sup n→∞ |kn

P(N_rn,n∈M)−n[P(N_m,n∈M)−P(N_m−1,n∈M)]|=0,

for any set M=∩d_i=1{µ∈Mp(E);µ(Bi)≥li}, with Bi∈ B, li≥1(integers) and d≥1.

Proof: Leth:Rd

+→R+ be a twice continuously differentiable function which satisfies (4), such

thath(x₁, . . . ,xd)≤x1+. . .+xdfor all(x1, . . . ,xd)∈R+d, and

h(x₁, . . . ,x_d) =

0 ifxi≤li−1 for somei=1, . . . ,d 1 ifxi≥lifor alli=1, . . . ,d Note that:

h(x₁, . . . ,x_d) =1{x1≥l1,...,xd≥ld} for allx1, . . . ,xd∈Z+. (8)

For anyn≥1, we consider strictly stationary sequence ofd-dimensional random vectors{Y_j,n= (Y_j(_,1_n), . . . ,Y_j(_,nd)), 1≤ j≤n}defined by:

Using (8), we obtain for any 1≤m≤n,

P(Nm,n∈M) =P(Nm,n(B1)≥l1, . . . ,Nm,n(Bd)≥ld) =

P(

m

X

j=1

Y_j(_,1_n)≥l1, . . . ,

m

X

j=1

Y_j(_,d_n)≥ld) =E[1{Pm j=1Y

(1)

j,n≥l1,...,

Pm j=1Y

(d)

j,n≥ld}] =

E[h(

m

X

j=1

Y_j(_,1_n), . . . , m

X

j=1

Y_j(_,d_n))] =E[h(

m

X

j=1 Y_j,n)].

Using Lemma 3.3, and lettingC=kD2hk∞, we obtain:

kn|P(Nrn,n∈M)−rn[P(Nm,n∈M)−P(Nm−1,n∈M)]| ≤ mkn{E[h(

m

X

j=1

Y_j,n)] +E[h(

m−1 X

j=1

Y_j,n)]}+C kn d

X

i,l=1

rn−m X

k=0 E(

k

X

j=1

Y_j(_,i_n)Y_k(₊l)_m,n)

=:I_m(1_,)_n+C I_m(2_,)_n (9)

Using the fact thath(x)≤Pd_i=1x_i, and the stationary of(X_j,n)j≤n,

I(_m1_,)_n ≤ 2mk_nE(

m

X

j=1

d

X

i=1

Y_j(_,i_n)) =2mk_n

d

X

i=1

m

X

j=1

P(X_j,n∈B_i)

= 2m2kn d

X

i=1

P(X1,n∈Bi)≤2m2 1

rn d

X

i=1

nP(X1,n∈Bi).

From (6), it follows that limn→∞Im(1,)n=0 for allm, and hence lim

m→m0

lim sup n→∞

I_m(1_,)_n=0. (10)

Using the stationarity of(X_j,n)_j≤n, and lettingB=∪d

i=1Bi∈ B,

I(_m2_,)_n = kn d

X

i,l=1

rn−m X

k=0

k

X

j=1

P(X_j,n∈Bi,Xk+m,n∈Bl)

= k_n

d

X

i,l=1

rn X

j=m+1

(r_n−j+1)P(X_1,n∈B_i,X_j,n∈B_l)

≤ n

d

X

i,l=1

rn X

j=m+1

P(X_1,n∈Bi,Xj,n∈Bl)

≤ d2n

rn X

j=m+1

P(X_1,n∈B,Xj,n∈B).

From (7), it follows that:

lim m→m0

lim sup n→∞

From, (9), (10) and (11), it follows that: lim

m→m0

lim sup n→∞

k_n|P(N_rn,n∈M)−r_n[P(N_m,n∈M)−P(N_m−1,n∈M)]|=0. Note that lim_n→∞(n−k_nr_n)|P(N_m,n∈M)−P(N_m−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.ƒ

Corollary 3.5. For each n≥1, let(X_j,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 N_m,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→∞ |

k_nP(N_rn,n∈M,M_rn,n>x)−n[P(N_m,n∈M,M_m,n>x)−

P(N_m−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.ƒ

Proof of Theorem 2.5:The result follows from Theorem 2.4 and Corollary 3.5.ƒ

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(X_j)_j≥1be a strictly stationary sequence of random variables. The extremal indexof the sequence (X_j)_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].

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[18].

Definition 4.3. We say that(X_j)_j≥1satisfiescondition (AIM)(or admits an asymptotic indepen-dence representation for the maximum) if there exists(r_n)_n⊂N_{withrn→ ∞andkn}= [n/r_n]→ ∞, 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(ξ_j)_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(X_j)_j≥1be a strictly stationary sequence whose extremal indexθexists, and(un)n

be a sequence of real numbers satisfying (12). Suppose that(X_j)_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>u_n)→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 (Y_i)_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.

Proposition 4.7. Let(X_j)_j≥1be a strictly stationary sequence whose extremal index θ exists, and (u_n)_nbe a sequence of real numbers satisfying (12). If(X_j)_j≥1satisfies (AIM), then knP(maxj≤rnXj> un)→θ.

Proof:Due to (AIM),P(maxj≤rnXj≤un)

kn→_e−θ_{. The result follows, since}

P(max j≤rn

X_j≤u_n)kn₌ ‚

1−knP(maxj≤rnXj>un) kn

Œkn

.

ƒ

Proposition 4.8. Let(Xj)j≥1be a strictly stationary sequence such that:

lim sup n→∞

nP(X₁>u_n)<∞.

Suppose that there exists(r_n)_n⊂N_{with rn→ ∞and kn}= [n/r_n]→ ∞, 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.ƒ

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).

ƒ

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].

References

[1] R. M. Balan and S. Louhichi, S. Convergence of point processes with weakly dependent points.J. Theoret. Probab.22(2009), 955-982. MR2558660. MR2558660

[2] R. M. Balan and S. Louhichi. A cluster limit theorem for infinitely divisible point processes (2009). Preprint available at: http://arxiv.org/abs/0911.5471. Math. Review number not available.

[3] A. Balkema and S. I. Resnick. Max-infinite divisibility.J. Appl. Probab.14(1977), 309-319. MR0438425. MR0438425

[4] K. Bartkiewicz, A. Jakubowski, T. Mikosch and O. Wintenberger. Stable limits for sums of dependent infinite variance random variables. To appear inProbab. Th. Rel. Fields (2009). Math. Review number not available.

[5] J. Beirlant, Y. Goegebeur, Y, J. Segers and J. TeugelsStatistics of Extremes: Theory and Appli-cations(2004). Wiley Series in Probability and Statistics. MR2108013. MR2108013

[6] J. Dedecker and S. Louhichi. Conditional convergence to infinitely divisible distributions with finite variance.Stoch. Proc. Appl.115(2005), 737-768. MR2132596. MR2132596 [7] J. Dedecker and S. Louhichi. Convergence to infinitely divisible distributions with finite

vari-ance.ESAIM Probab. Stat.9(2005), 38-73. MR2148960. MR2148960

[8] R. A. Davis and T. Hsing. Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23(1995), 879-917. MR1334176. MR1334176

[9] R. Durrett and S. I. Resnick. Functional limit theorems for dependent variables.Ann. Probab. 6(1978), 829-846. MR0503954. MR0503954

[10] B. V. Gnedenko and A. N. Kolmogorov.Limit Distributions for Sums of Independent Random Variables(1954). Addison-Wesley, MA. MR0062975. MR0062975

[11] T. Hsing, J. Hüsler and M. R. Leadbetter. On the exceedance process for a stationary se-quence.Probab. Th. Rel. Fields78(1988), 97-112. MR0940870. MR0940870

[12] A. Jakubowski. Minimal conditions inp-stable limit theorems.Stoch. Proc. Appl.44(1993), 291-327. MR1200412. MR1200412

[13] A. Jakubowski. Minimal conditions inp-stable limit theorems.Stoch. Proc. Appl.68(1997), 1-20. MR1454576. MR1454576

[14] A. Jakubowski and M. Kobus.α-stable limit theorems for sums of dependent random vectors.

J. Multiv. Anal.29(1989), 219-251. MR1004336. MR1004336

[15] O. Kallenberg.Random Measures. Third edition (1983). Springer, New York. MR0818219. MR0818219

[16] M. R. Leadbetter. Extremes and local dependence in stationary sequences. Z. Wahr. verw. Gebiete65(1983), 291-306. MR0722133. MR0722133

[17] G. F. Newell. Asymptotic extremes form-dependent random variables.Ann. Math. Stat.35

(1964), 1322-1325. MR0164361. MR0164361

[18] G. L. O’Brien. The maximal term of uniformly mixing stationary processes.Z. Wahr. verw. Gebiete30(1974), 57-63. MR0362451. MR0362451

[19] G. L. O’Brien. Extreme values for stationary and Markov sequences.Ann. Probab.15(1987), 281-291. MR0877604. MR0877604

[20] S. I. Resnick. Weak convergence to extremal processes.Ann. Probab. 3 (1975), 951-960. MR0428396. MR0428396

[21] S. I. Resnick.Extreme Values, Regular Variation, and Point Processes(1987). Springer, New York. MR0900810. MR0900810

[22] S. I. Resnick.Heavy Tail Phenomena: Probabilistic and Statistical Modelling(2007). Springer, New York. MR2271424. MR2271424

[23] S. I. Resnick and P. Greenwood. A bivariate stable characterization and domains of attraction.

J. Multiv. Anal.9(1979), 206-221. MR0538402. MR0538402

[24] J. Segers. Approximate Distributions of Clusters of Extremes.Statist. Probab. Lett.74(2005), 330-336. MR2186477. MR2186477

[25] J. Segers. Rare events, temporal dependence, and the extremal index.J. Appl. Probab.43

(2006), 463-485. MR2248577. MR2248577

[26] R. L. Smith. The extremal index for a Markov chain. J. Appl. Probab.29 (1992), 37-45. MR1147765. MR1147765

[27] I. Weissman. On weak convergence of extremal processes.Ann. Probab.4(1976), 470-473. MR0400330. MR0400330

Using (8), we obtain for any 1≤m≤n,

P(Nm,n∈M) =P(Nm,n(B1)≥l1, . . . ,Nm,n(Bd)≥ld) = P(

m

X

j=1

Y_j(_,1_n)≥l1, . . . ,

m

X

j=1

Y_j(_,d_n)≥ld) =E[1{Pmj=1Y

(1)

j,n≥l1,...,

Pm j=1Y

(d)

j,n≥ld}] = E[h(

m

X

j=1

Y_j(_,1_n), . . . ,

m

X

j=1

Y_j(_,d_n))] =E[h(

m

X

j=1 Y_j,n)].

Using Lemma 3.3, and lettingC=kD2hk∞, we obtain:

kn|P(Nrn,n∈M)−rn[P(Nm,n∈M)−P(Nm−1,n∈M)]| ≤ mkn{E[h(

m

X

j=1

Y_j,n)] +E[h(

m−1 X

j=1

Y_j,n)]}+C kn d

X

i,l=1

rn−m

X

k=0

E(

k

X

j=1

Y_j(_,i_n)Y_k(₊l)_m,n)

=:I_m(1_,)_n+C I_m(2_,)_n (9)

Using the fact thath(x)≤Pd_i=1x_i, and the stationary of(X_j,n)j≤n,

I(_m1_,)_n ≤ 2mk_nE(

m

X

j=1

d

X

i=1

Y_j(_,i_n)) =2mk_n d

X

i=1

m

X

j=1

P(X_j,n∈B_i)

= 2m2kn d

X

i=1

P(X1,n∈Bi)≤2m2

1

rn d

X

i=1

nP(X1,n∈Bi).

From (6), it follows that limn→∞Im(1,)n=0 for allm, and hence

lim

m→m0

lim sup

n→∞

I_m(1_,)_n=0. (10)

Using the stationarity of(X_j,n)_j≤n, and lettingB=∪d

i=1Bi∈ B,

I(_m2_,)_n = kn d

X

i,l=1

rn−m

X

k=0

k

X

j=1

P(Xj,n∈Bi,Xk+m,n∈Bl)

= k_n d

X

i,l=1

rn

X

j=m+1

(r_n−j+1)P(X_1,n∈B_i,X_j,n∈B_l)

≤ n

d

X

i,l=1

rn

X

j=m+1

P(X1,n∈Bi,Xj,n∈Bl)

≤ d2n rn

X

j=m+1

P(X1,n∈B,Xj,n∈B).

From (7), it follows that:

lim

m→m0

lim sup

n→∞

From, (9), (10) and (11), it follows that: lim

m→m0

lim sup

n→∞

k_n|P(N_rn,n∈M)−r_n[P(N_m,n∈M)−P(N_m−1,n∈M)]|=0.

Note that lim_n→∞(n−k_nr_n)|P(N_m,n∈M)−P(N_m−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.ƒ

Corollary 3.5. For each n≥1, let(X_j,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 N_m,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→∞ |

k_nP(N_rn,n∈M,M_rn,n>x)−n[P(N_m,n∈M,M_m,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.ƒ

Proof of Theorem 2.5:The result follows from Theorem 2.4 and Corollary 3.5.ƒ

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(X_j)_j≥1be a strictly stationary sequence of random variables. The extremal indexof the sequence (X_j)_j≥1, if it exists, is a real numberθ with the following property: for anyτ >0, there exists a sequence(u(_nτ))_n⊂R_{such 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].

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[18].

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(ξ_j)_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(X_j)_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>u_n)→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 (Y_i)_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.

Proposition 4.7. Let(Xj)j≥1be a strictly stationary sequence whose extremal index θ exists, and (un)nbe a sequence of real numbers satisfying (12). If(Xj)j≥1satisfies (AIM), then knP(maxj≤rnXj> un)→θ.

Proof:Due to (AIM),P(max_j≤r

nXj≤un)

kn→_e−θ_{. The result follows, since}

P(max

j≤rn

X_j≤u_n)kn₌

‚

1−knP(maxj≤rnXj>un) kn

Œkn

.

ƒ

Proposition 4.8. Let(Xj)j≥1be a strictly stationary sequence such that: lim sup

n→∞

nP(X₁>u_n)<∞.

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(jmax≤m−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.ƒ

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).

ƒ

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].

References

[1] R. M. Balan and S. Louhichi, S. Convergence of point processes with weakly dependent points.J. Theoret. Probab.22(2009), 955-982. MR2558660. MR2558660

[2] R. M. Balan and S. Louhichi. A cluster limit theorem for infinitely divisible point processes (2009). Preprint available at: http://arxiv.org/abs/0911.5471. Math. Review number not available.

[3] A. Balkema and S. I. Resnick. Max-infinite divisibility.J. Appl. Probab.14(1977), 309-319. MR0438425. MR0438425

[4] K. Bartkiewicz, A. Jakubowski, T. Mikosch and O. Wintenberger. Stable limits for sums of dependent infinite variance random variables. To appear inProbab. Th. Rel. Fields (2009). Math. Review number not available.

[5] J. Beirlant, Y. Goegebeur, Y, J. Segers and J. TeugelsStatistics of Extremes: Theory and Appli-cations(2004). Wiley Series in Probability and Statistics. MR2108013. MR2108013

[6] J. Dedecker and S. Louhichi. Conditional convergence to infinitely divisible distributions with finite variance.Stoch. Proc. Appl.115(2005), 737-768. MR2132596. MR2132596

[7] J. Dedecker and S. Louhichi. Convergence to infinitely divisible distributions with finite vari-ance.ESAIM Probab. Stat.9(2005), 38-73. MR2148960. MR2148960

[8] R. A. Davis and T. Hsing. Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23(1995), 879-917. MR1334176. MR1334176

[9] R. Durrett and S. I. Resnick. Functional limit theorems for dependent variables.Ann. Probab.

6(1978), 829-846. MR0503954. MR0503954

[10] B. V. Gnedenko and A. N. Kolmogorov.Limit Distributions for Sums of Independent Random Variables(1954). Addison-Wesley, MA. MR0062975. MR0062975

[11] T. Hsing, J. Hüsler and M. R. Leadbetter. On the exceedance process for a stationary se-quence.Probab. Th. Rel. Fields78(1988), 97-112. MR0940870. MR0940870

[12] A. Jakubowski. Minimal conditions inp-stable limit theorems.Stoch. Proc. Appl.44(1993), 291-327. MR1200412. MR1200412

[13] A. Jakubowski. Minimal conditions inp-stable limit theorems.Stoch. Proc. Appl.68(1997), 1-20. MR1454576. MR1454576

[14] A. Jakubowski and M. Kobus.α-stable limit theorems for sums of dependent random vectors.

J. Multiv. Anal.29(1989), 219-251. MR1004336. MR1004336

[15] O. Kallenberg.Random Measures. Third edition (1983). Springer, New York. MR0818219. MR0818219

[16] M. R. Leadbetter. Extremes and local dependence in stationary sequences. Z. Wahr. verw. Gebiete65(1983), 291-306. MR0722133. MR0722133

[17] G. F. Newell. Asymptotic extremes form-dependent random variables.Ann. Math. Stat.35 (1964), 1322-1325. MR0164361. MR0164361

[18] G. L. O’Brien. The maximal term of uniformly mixing stationary processes.Z. Wahr. verw. Gebiete30(1974), 57-63. MR0362451. MR0362451

[19] G. L. O’Brien. Extreme values for stationary and Markov sequences.Ann. Probab.15(1987), 281-291. MR0877604. MR0877604

[20] S. I. Resnick. Weak convergence to extremal processes.Ann. Probab. 3 (1975), 951-960. MR0428396. MR0428396

[21] S. I. Resnick.Extreme Values, Regular Variation, and Point Processes(1987). Springer, New York. MR0900810. MR0900810

[22] S. I. Resnick.Heavy Tail Phenomena: Probabilistic and Statistical Modelling(2007). Springer, New York. MR2271424. MR2271424

[23] S. I. Resnick and P. Greenwood. A bivariate stable characterization and domains of attraction.

J. Multiv. Anal.9(1979), 206-221. MR0538402. MR0538402

[24] J. Segers. Approximate Distributions of Clusters of Extremes.Statist. Probab. Lett.74(2005), 330-336. MR2186477. MR2186477

[25] J. Segers. Rare events, temporal dependence, and the extremal index.J. Appl. Probab.43 (2006), 463-485. MR2248577. MR2248577

[26] R. L. Smith. The extremal index for a Markov chain. J. Appl. Probab.29 (1992), 37-45. MR1147765. MR1147765

[27] I. Weissman. On weak convergence of extremal processes.Ann. Probab.4(1976), 470-473. MR0400330. MR0400330