The Borel–Cantelli lemma under dependence conditions

Statistics & Probability Letters 78 (2008) 390–395

The Borel–Cantelli lemma under dependence conditions

Tapas Kumar Chandra

BIRU, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India Received 9 October 2006; received in revised form 3 April 2007; accepted 25 July 2007

Available online 25 August 2007

Abstract

Generalizations of the second Borel–Cantelli lemma are obtained under very weak dependence conditions, subsuming several earlier results as special cases.

r2007 Elsevier B.V. All rights reserved.

Keywords:Borel–Cantelli lemma; Generalizations of the Borel–Cantelli lemma; Pairwise independence; Pairwisem-dependence; Pairwise negative quadrant dependence

1. Introduction

In a recent note, Petrov (2004) proved using clever arguments an interesting extension of the (second) Borel–Cantelli lemma; the theorem in Section 2 of Petrov (2004) contains several earlier extensions of the Borel–Cantelli lemma as special cases. This note extends Petrov’s result further (see Theorems 1 and 1⁰); it is expected that the present version will be more widely applicable. A related result is also included in the final section.

To motivate the extension, we state the following results. If there are non-negative realsc1, c2,c3 and an integerNX1 such that

PðAi\AjÞpðc1PðAiÞ þc2PðAjÞÞPðAjiÞ þc3PðAiÞPðAjÞ (1) wheneverNpioj, and

X¹

n¼1

PðA_nÞ ¼ 1, (2)

thencX1 and

Pðlim supAnÞX1

c, (3)

where

c:¼c₃þ2ðc₁þc₂Þ. (4)

www.elsevier.com/locate/stapro

0167-7152/$ - see front matterr2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.spl.2007.07.023

E-mail address:tkchandra@gmail.com

The result of Petrov (2002) is a special case where c₁¼c₂¼0. For a simple proof, see Chandra (1999).

Regarding condition (1), see Kochen and Stone (1964, Example 2) andLamperti (1963, p. 62). On the other hand, the extended Re´nyi–Lamperti lemma states that if

lim inf

n!1

P1pj;kpnPðAj\AkÞ ðP

1pjpnPðAjÞÞ² ¼c (5)

and (2) holds, thencX1 and (3) holds. (The Re´nyi–Lamperti lemma states that Pðlim supAnÞX2c;

seeBillingsley, 1979, p. 76.) For a simple proof, seeSpitzer (1964, p. 319).Erdo¨s and Re´nyi (1959)considers the special case where (5) holds withc¼1, and contains the case where (1) holds withc₁¼c₂¼0 andc₃¼1 and (2) obtains; in particular, the second Borel–Cantelli lemma holds if the independence offAngis replaced by pairwise independence or pairwise negative quadrant dependence of fAng(this fact can also be obtained from the result that (1) and (2) together imply (3)). In his paper,Petrov (2004)combined conditions (1), with c1¼c2 ¼0; and (5) in a clever manner and obtained a common generalization of Petrov (2002) and the extended Re´nyi–Lamperti lemma; see, also,Ortega and Wschebor (1983). He used the following inequality of Chung and Erdo¨s (1952):

P [ⁿ

k¼1

A_k

!

X ðPn

k¼1PðAkÞÞ² Pn

j;k¼1PðA_j\A_kÞ, (6)

which is, in turn, a special case of Spitzer’s inequality (seeSpitzer, 1964, p 319): IfEðXÞ40 and 0po1, then PðX4EðXÞÞXð1Þ²ðEðXÞÞ²

EðX²Þ . (7)

(Take ¼0 and X¼Pn

k¼1IAk.) We shall prove the following extension of Petrov (2004) using suitable modifications of his arguments.

Theorem 1. LetfA_ng_nX1 satisfy(2) and assume that lim inf

n!1

P1pjokpnðPðAj\AkÞ ajkÞ ðP

1pkpnPðA_kÞÞ² ¼L,

where a_ij¼ ðc₁PðA_iÞ þc₂PðA_jÞÞPðA_jiÞ þc₃PðA_iÞPðA_jÞ for 1pioj, c₁X0, c₂X0, c₃ being constants (L may depend on c₁,c₂ and c₃).Assume that L is finite. Then cþ2LX1and

Pðlim supAnÞX 1 ðcþ2LÞ, where c is given by(4).

Petrov’s result follows whenc1 ¼c2¼0. Furthermore, it follows that (1) and (2) imply (3), since thenLp0 and soðcþ2LÞ¹Xc¹:

The proof of Theorem 1 shows that the following more general result is also true; the details are omitted.

Theorem 1⁰. Let fAng_nX1 satisfy (2)and lim inf

n!1

P

1pjokpnðPðA_j\A_kÞ a_jkÞ ðP

1pkpnPðAkÞÞ² ¼L, where the ajk satisfy

X

1pjompkpn

ja_jkj ¼o X

mpkpn

PðA_kÞ

!2

0

@

1

A 8mX1

and

lim inf

m!1 lim sup

n!1

Pmpjokpnajk

ðP

mpkpnPðA_kÞÞ²pd.

Assume that L and d are finite. Then dþLX¹

2and Pðlim supA_nÞXð2dþ2LÞ¹: 2. Proof of Theorem 1

We need the following lemma.

Lemma 1. Let(2) hold and a_ij be as in Theorem1.Put b_m¼lim inf

n!1

PmpjokpnðPðAj\AkÞ ajkÞ ðPn

k¼mPðA_kÞÞ² ; mX1.

Thenb₁ ¼b_m,8mX1.

Proof. Fix an integerm41. As (2) holds, we have b₁¼ lim inf

n!1

P

1pjokpnðPðA_j\A_kÞ a_jkÞ ðPn

k¼mPðAkÞÞ²

¼ lim inf

n!1 ðI1;nþI2;nþI3;nÞ, where, fornXm,

sm;n¼Xⁿ

k¼m

PðAkÞ; bj;k¼PðAj\AkÞ ajk,

I_1;n¼

P1pjokpmb_j;k

s²_m;n ; I_2;n¼

Pmpjokpnb_j;k s²_m;n , I_3;n¼

P1pjompkpnbj;k

s²_m;n .

Condition (2) implies thatI1;n!0 asn! 1. Also jI3;njp

P

1pjompkpnðPðA_j\A_kÞ þ ja_jkjÞ s²_m;n

pmð1þ jc₁j þ jc₂j þ jc₃jÞ sm;n

þms_1;mjc₁j s²_m;n so thatI_3;n !0 asn! 1. Therefore,

b₁¼b_mþlimI1;nþlimI3;n ¼b_m: &

We next prove Theorem 1. LetmX1 be an integer; letNXmbe such thatPN

k¼mPðA_kÞ40. Then, by (6), P [¹

k¼m

Ak

!

X ðPn

k¼mPðA_kÞÞ² Pn

j;k¼mPðAj\AkÞ 8nXN.

Now, Xⁿ

j;k¼m

PðAj\AkÞ ¼sm;nþT1þT2,

where thes_m;n are as in the proof of Lemma 1 and T₁ ¼2 X

mpjokpn

ðPðA_j\A_kÞ a_jkÞ,

T₂ ¼2 X

mpjokpn

a_jk,

where theajk are as in the statement of Theorem 1. As T2p2 c3

2 þc1þc2

s²_m;nþ ðc1þc2Þs1;msm;nc3

Xⁿ

j¼m

ðPðAjÞÞ², one has

lim sup

n!1

T₂ s²_m;npc,

wherecis given by (4). Therefore, condition (2) implies that P [¹

k¼m

A_k

!

X lim inf

n!1

T₁

s²_m;nþlim sup

n!1

T₂ s²_m;n

( )1

Xf2Lþcg¹ by Lemma 1.

3. An alternative approach

In this section we derive another version of the second Borel–Cantelli lemma under a suitable dependence condition using the Chebyshev Inequality.

Lemma 2. Let fXngbe a sequence of non-negative random variables with finite EðX²_nÞand put Sn ¼Pn i¼1Xi, nX1.Assume that EðSnÞ ! 1as n! 1,and

VarðSnÞpcnEðSnÞ 8nX1; lim inf

n!1

cn

EðS_nÞ

¼0. (8)

Then P X¹

n¼1

Xn¼ 1

!

¼1. (9)

Proof. Note that P X¹

n¼1

Xno1

!

¼ lim

n!1P X¹

n¼1

Xnp1 2EðSnÞ

!

plim inf

n!1 P Snp1 2EðSnÞ

plim inf

n!1

4c_n EðSnÞ

¼0: &

Theorem 2. LetfXngbe a sequence of non-negative random variables with finite EðX²_nÞand put Sn ¼Pn i¼1Xi, nX1.Assume that EðSnÞ ! 1as n! 1.

(a)Assume that Xⁿ

i¼1

VarðXiÞpknEðSnÞ 8nX1, (10)

Xⁿ

j¼2

X^j1

i¼1

CovðX_i;X_jÞpc_nEðS_nÞ 8nX2 (11)

and

lim inf

n!1 ððk_nþ2c_nÞ=EðS_nÞÞ ¼0. (12)

Then(9) holds.If0pXnpkn 8nX1wherefkngis nondecreasing,then (10)holds.

(b) If0pXnpkn 8nX1 wherefkng is nondecreasing, and fqðmÞg, famgand fbmgare non-negative sequences anda,bare non-negative constants such that

CovðXi;XjÞpqðjijjÞðaiþbjÞ 8iaj, (13)

Xⁿ¹

i¼1

aipaEðSnÞ; Xⁿ

j¼2

bjpbEðSnÞ 8nX2 (14)

and

lim inf

n!1

Pn1 m¼1qðmÞ EðS_nÞ

!

¼0; lim

n!1ðkn=EðSnÞÞ ¼0, (15)

then(9) holds.

Proof. (a) follows from Lemma 2.

(b) follows from Part (a) and the following observation:

Xⁿ

j¼2

X^j1

i¼1

CovðX_i;X_jÞpXⁿ

j¼2

X^j1

i¼1

qðjiÞða_iþb_jÞ

¼ Xⁿ

j¼2

X^j1

m¼1

qðmÞðajmþbjÞ

¼ Xⁿ¹

m¼1

qðmÞ Xⁿ

j¼mþ1

ðajmþbjÞ

pðaþbÞ Xⁿ¹

m¼1

qðmÞ

!

EðSnÞ: &

We remark that one choice of thea_m andb_m so that (14) holds with appropriatea andbis a_m¼EðX_mÞ þEðX_mþ1Þ; b_m¼EðX_mÞ þEðX_m1Þ.

Remark 1. If in the above theorem, we put X_n¼I_An, k_n ¼1 8nX1 so that lim supA_n¼ ½P1

n¼1X_n¼ 1 wherefA_ngis a given sequence of events satisfyingP1

n¼1PðA_nÞ ¼ 1, we get another set of sufficient conditions forPðlim supA_nÞ ¼1. Thus, we have: If (2) holds, and for eachioj,

PðA_i\A_jÞ PðA_iÞPðA_jÞpqðjijjÞ½PðA_iÞ þPðA_iþ1Þ þPðA_jÞ þPðA_j1Þ,

lim inf

n!1

Pn1 m¼1qðmÞ Pn

m¼1PðAmÞ

!

¼0 a fortioriX¹

m¼1

qðmÞo1

! ,

thenPðlim supA_nÞ ¼1:In particular, if there exists an integermX0 such that

PðAi\AjÞpPðAiÞPðAjÞ wheneverjijj4m (16)

and (2) holds, then Pðlim supAnÞ ¼1. The last inequality definitely holds if IAi and IAj are pairwise m-dependent (a fortiori, pairwise independent).

Remark 2. LetfXngbe as in Theorem 2(b). Suppose that the conditions of Theorem 2(b) hold withkn¼1, 8nX1, except that (13) and (15) are strengthened to

CovðX_i;X_jÞpqðjjijÞEðX_jÞ 8ioj, (17)

X¹

n¼1

ðqðnÞ=EðS_nÞÞo1. (18)

Then (9) can be strengthened to ‘S_n=EðS_nÞ !1 a.s.’ which gives an indication of the rate of growth of S_n in (9). For a proof, see Chandra and Ghosal (1998). Note that if (16) holds for some mX0, then (17) and (18) hold with an appropriateq.

It is an interesting problem to derive the best possible result in the setup of either of the two remarks.

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