Maximal Inequalities for Dependent Random Variables and Applications

(1)

Volume 2008, Article ID 598319,10pages doi:10.1155/2008/598319

Research Article

Maximal Inequalities for Dependent Random

Variables and Applications

Soo Hak Sung

Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea

Correspondence should be addressed to Soo Hak Sung,[email protected]

Received 16 April 2008; Revised 3 June 2008; Accepted 7 July 2008

Recommended by Ondrej Dosly

For a sequence {Xn, n ≥ 1} of dependent square integrable random variables and a sequence

{bn, n≥1}of positive numbers, we establish a maximal inequality for weighted sums of dependent random variables. Applying this inequality, we obtain the almost sure convergence of ni1Xi/bi andni1Xi/bn.

Copyrightq2008 Soo Hak Sung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Throughout this paper let {Xn, n ≥ 1} be a sequence of random variables defined on a

probability spaceΩ,F, Pand let{bn, n ≥ 1}be a sequence of positive numbers. We assume

that there exists a sequence{ρn, n≥1}of nonnegative constants such that

sup

k≥1

EXkXkn≤ρn, forn≥1. 1.1

In this paper, we establish a maximal inequality for weighted sums of the dependent random variables satisfying1.1. Applying this inequality, we obtain under some suitable conditions on the sequence{ρn}that

n

i1 Xi

bi

converges a.s. asn−→ ∞ 1.2

and the strong law of large numbersSLLN

_n

i1Xi

bn −→0 a.s. 1.3

(2)

For a sequence of dependent random variables satisfying 1.1, the SLLNs were established by Hu et al.1,2and Lyons3. Lyons3obtained an SLLN under the conditions

that VarXn O1and bn n. Without condition VarXn O1, Hu et al. 1 obtained

an SLLN, wherebn n.Hu et al.2also obtained an SLLN for more general sequence{bn}

bnnis replaced bynObn.

For other results on the SLLN for a sequence of correlated random variables, see Chandra

4, M ´oricz5,6, and Serfling7,8.

In this paper, we give a suﬃcient condition under which1.2and1.3hold. Our results

partially improve those of Hu et al. 1, 2. The technique used in our proof is the well-known method of subsequences. Note that the maximal inequality is used in the method of subsequences. Our maximal inequality for weighted sums of the dependent random variables satisfying1.1is sharper than that of Hu et al.2.

Throughout this paper, logxdenotes the natural logarithm.

2. Maximal inequalities for dependent random variables

To prove the maximal inequality for weighted sums of dependent random variables satisfying

1.1, the following lemma is needed.

Lemma 2.1. Let{Xn, n≥1}be a sequence of square integrable random variables satisfying1.1. Let

{bn, n≥1}be a sequence of positive numbers such that

n≤Dbn ∀n≥1and some constantD > 0. 2.1

Then for alln≥1, m > n,andδ >0,

m−1

in m

ji1

EXiXj

bibj ≤

D2_C

δ

max{log 2δ,lognδ}

m−n

k1 ρk

k 1logk

1δ_, _2.2

whereCδ2δ1max{1, δδe−δ}.

Proof. For simplicity of notation, letIn,m

_m₋₁

in

_m

ji1EXiXj/bibj.Then we get by 1.1

and2.1that for 1≤n < m,

In,m≤ m−1

in m

ji1 ρj−i

bibj

≤D2

m−1

in m

ji1 ρj−i

ij D2

m−n

k1

m−k

in

ρk

iik D2

m−n

k1 ρk

k

m−k

in

1

i −

1

ik

≤D2

m−n

k1 ρk

k

nk−1

in

1

i.

(3)

We next estimate_in_nk−11/i.Ifn1,then

nk−1

in

1

i

k

i1

1

i ≤1

k

1

xdx≤1logk≤1logk

1δ_. _2.4

Ifn≥2,then

nk−1

in

1

i ≤

nk−1

n−1

1

xdxlog

1 k

n−1

≤2 log

1k

n

. 2.5

The log1k/nis estimated as follows:

log

1 k

n ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ log

1 √1

n

≤ logn

δ

lognδ√n ≤

2δδe−δ

lognδ ≤2δ

δ_e−δ1logk

1δ

lognδ , if 1≤k≤ √

n,

log

1k

n

_{2 log}_kδ

lognδ ≤2

δlogk

1δ

lognδ ≤2

δ1logk

1δ

lognδ , if k > √

n. 2.6

Thus, we have the desired estimate forIn,m:

In,m≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D2m−n

k1 ρk

k 1logk

1δ_, _if _n₁_,

D2

m−n

k1 ρk

k

2 max{2δ,2δδe−δ}

lognδ 1logk

1δ_, _if _n_≥₂_,

≤ D22δ1max{1, δδe−δ}

max{log 2δ,lognδ}

m−n

k1 ρk

k 1logk 1δ_.

2.7

The following lemma is a maximal inequality for general dependent random variables.

Lemma 2.2. Let{Xn, n ≥ 1}be a sequence of square integrable random variables. Then for alla ≥ 0

andn≥1,

E

max

1≤k≤n

ak

ia1 Xi 2 ≤ log 2n

log 2

2an

ia1

EX_i22

an−1

ia1

an

ji1

EXiXj

. 2.8

Proof. LetFa,nbe the joint distribution function ofXa1, . . . , Xan.Define a functiongon{Fa,n:

a≥0, n≥1}by

gFa,n

an

ia1

EX2_i 2

an−1

ia1

an

ji1

(4)

Then we can easily obtain that fora≥0, k≥1,andm≥1,

gFa,k gFak,m≤gFa,km. 2.10

Moreover, we have that for alla≥0 andn≥1,

E

_a_n

ia1 Xi

2

≤gFa,n. 2.11

By Serfling’s9generalization of the Rademacher-Menchoﬀmaximal inequality for orthogo-nal random variables,

E

max

1≤k≤n

ak

ia1 Xi

2 ≤

_{log 2}_n log 2

2

gFa,n. 2.12

Thus, the result is proved.

Combining Lemmas 2.1 and 2.2gives the following maximal inequality for weighted

sums of dependent random variables satisfying1.1.

Lemma 2.3. Let{Xn, n≥1}be a sequence of square integrable random variables satisfying1.1. Let

{bn, n≥1}be a sequence of positive numbers satisfying2.1. Then for alln≥1, m > n,andδ >0,

E

max

n≤i≤m

i

jn

Xj

bj

2

≤

log2m−n1 log 2

2m

in

EX_i2 b2_i

2D2Cδ

max{log 2δ,lognδ}

m−n

k1 ρk

k 1logk 1δ

, 2.13

whereCδ2δ1max{1, δδe−δ}.

3. Almost surely convergent series and strong laws of large numbers

In this section, we will assume that {Xn, n ≥ 1} is a sequence of square integrable random

variables satisfying1.1. A suﬃcient condition will be given under which1.2and1.3hold. We first state and prove one of our main results. The proof is based on the well known method of subsequences. Our proof is similar to that of Hu et al. 2. However, the maximal inequalityLemma 2.3used in the proof is sharper than that of Hu et al.2.

Theorem 3.1. Let{Xn, n≥1}be a sequence of square integrable random variables satisfying1.1. Let

{bn, n ≥ 1}be a sequence of positive numbers satisfying2.1. Suppose that the following conditions

hold:

i∞_n₁logn2EX2

n/b2n<∞,

ii∞n1logn4δρn/n <∞for someδ >0.

(5)

Proof. As noted in the introduction, if 0 < bn ↑ ∞,then1.2implies1.3. To prove 1.2, let

Snni1Xi/bi.ByLemma 2.1withδreplaced by 3δ, we have that form > n,

ESm−Sn2 m

in1 EX2_i

b2

i

2

m−1

in1

m

ji1 EXiXj

bibj

≤ m

in1 EX2_i

b2

i

2D2C3δ

max{log 23δ,logn3δ}

m−n−1

k1 ρk

k1logk 4δ

≤ ∞

in1 EX2

i

b2

i

2D2C3δ

max{log 23δ,logn3δ}

∞

k1 ρk

k 1logk

4δ _−→₀

3.1

asn → ∞byiandii. HereC3δ2δ4max{1,3δ3δe−3δ}.By the Cauchy convergence

criterion, there exists a random variableSsuch thatESn−S2 → 0 asn → ∞.It is easy to see

thatS2n → Sa.s. by the standard method. It remains to show that

max

2n_<k_≤₂n1|Sk−S2

n| −→0 a.s. asn−→ ∞. 3.2

UsingLemma 2.3,i, andii, we get that

∞

n1

P max

2n_<k_≤₂n1|Sk−S2

n|>

≤ 1 2

∞

n1

E max

2n_<k_≤₂n1|Sk−S2

n|2

≤ 1 2

∞

n1

log 2n1 log 2

2 ₂n1

i2n₁ EX_i2

b2_i

2D2C3δ

log2n13δ

2n₋₁

k1 ρk

k1logk 4δ

≤ 1

2_{log 2}2

∞

i3

log2i2EX2

i

b2

i

2D2C3δ

2_{log 2}3δ

∞

n1

n12

n3δ

∞

k1 ρk

k1logk

4δ _<_∞_.

3.3

Then3.2follows by the Borel-Cantelli lemma.

Remark 3.2. Hu et al.2provedTheorem 3.1underiandii.

ii ∞_n₁ρn/nq<∞for some 0≤q <1.

Since conditioniiofTheorem 3.1is weaker thanii,Theorem 3.1improves the result of Hu et al.2.

We can now establish the following SLLN if condition2.1on{bn}is replaced by the

condition 0< bn↑ ∞.

{bn, n ≥ 1} be a nondecreasing unbounded sequence of positive numbers. Suppose that the following

conditions hold:

(6)

ii∞_n₁ρn∞in1logi2/b2i <∞,

iii∞_n₁EXn2

_∞

in1logi/ibi2<∞.

Then1.3holds.

To proveTheorem 3.3, we need the following lemma which is due to Fazekas and Klesov

10.

Lemma 3.4. Let {Xn, n ≥ 1}be a sequence of random variables and{bn, n ≥ 1}be a nondecreasing

unbounded sequence of positive numbers. Let {αn, n ≥ 1} be a sequence of nonnegative numbers.

Assume that for eachn≥1,

E

max

1≤i≤n

i

j1 Xj

r

≤n

i1

αi, for some constantr >0. 3.4

If∞_n₁αn/brn<∞,then1.3holds.

Proof ofTheorem 3.3. FromLemma 2.2,

E

max

1≤i≤n

i

j1 Xj 2 ≤ log 2n

log 2

2n

i1

EX_i22

n−1

i1

n

ji1

EXiXj

. 3.5

Defineαn log 2n/log 22An−log 2n−1/log 22An−1forn ≥ 1,where A0 0 andAn

_n

i1EX2i 2

_n₋₁

i1

_n

ji1EXiXjforn≥1.ThenEmax1≤i≤n|

_i

j1Xj|2≤

_n

i1αiand

αn

_{log 2}_n log 2

2

An−An−1 An−1

_{log 2}_n

log 2 2

−

_{log 2}_n₋₁ log 2

2

log 2n

log 2

2

EXn22 n−1

i1

EXiXn

log 2n

log 2 2

−

log 2n−1 log 2

2n−1

i1

EX2_i 2

n−2

i1

n−1

ji1

EXiXj

.

3.6

ByLemma 3.4, it is enough to show that ∞

n1

log 2n2EX2

n

b2

n

<∞, 3.7

∞

n1

log 2n2 b2n

n−1

i1

EXiXn<∞, 3.8

∞

n2

log 2n2−log 2n−12

b2n

n−1

i1 EX2

i <∞, 3.9

∞

n3

log 2n2−log 2n−12

b2

n

n−2

i1

n−1

ji1

EXiXj<∞. 3.10

(7)

The following corollary shows that conditioniiofTheorem 3.3can be simplified under the additional condition2.1on{bn}.

Corollary 3.5. Let{Xn, n ≥ 1}be a sequence of square integrable random variables satisfying1.1.

Let{bn, n≥1}be a nondecreasing unbounded sequence of positive numbers satisfying2.1. Suppose

that the following conditions hold: i∞_n₁logn2EX2

n/b2n<∞,

ii∞n1logn2ρn/n <∞,

iii∞_n₁EX2

n

_∞

in1logi/ibi2<∞.

Then1.3holds.

Proof. By2.1, we have that ∞

n1 ρn

∞

in1

logi2 b2

i

≤D2

∞

n1 ρn

∞

in1

logi2 i2 ≤C

∞

n1 ρn

logn2

n 3.11

for some constantC >0.Thus the result follows byTheorem 3.3.

Remark 3.6. ConditioniiofCorollary 3.5is weaker than conditioniiofTheorem 3.1. On the other hand, an additional condition is needed inCorollary 3.5namely conditioniiiabove.

Using the following lemma, we can omit condition iii of Theorem 3.3 if conditions

2.1 and3.12on {bn} are satisfied. If C1n ≤ bn ≤ C2nα for all n ≥ 1 and some constants

C1>0, C2>0,andα >0,then2.1and3.12hold.

Lemma 3.7. Let{bn, n ≥ 1}be a nondecreasing unbounded sequence of positive numbers satisfying

2.1. If

lim sup

n→ ∞

logbn

logn <∞, 3.12

then

b2

n

log2n

∞

in

logi

ib2_i O1. 3.13 Proof. Without loss of generality, we may assume thati≤bifor alli≥1.

Let fixn.For eachk ≥ 1,definemk bymk min{i ≥ n: bi ≥ kbn}.Thenbmk ≥ kbn and nm1≤m2≤ · · · .It follows that

b2n

log2n

∞

in

logi ib2

i

b2n

log2n

∞

k1

mk1−1

imk

logi ib2

i

≤ bn2

log2n

∞

k1

1

b2

mk

mk1−1

imk

logi

i by 0< bn ↑

≤ b2n

log2n

∞

k1

1

kbn2 mk1−1

imk

logi i

≤ 1

log2n

∞

k1

1

k2

3

i1

logi i

_m_k₁₋₁

3

logx x dx

≤ 1

log2n

∞

k1

1

k2

_{log 2}

2

log 3

3

logmk1−12

2

,

(8)

where we assume in the casemk1mk,the sum

mk1−1

imk 0.Sincebi≥ifor alli≥1, bkbn1≥ kbn 1≥kbnandmk ≤kbn 1≤kbn1.So we have that

logmk1−12≤logk1bn2≤2

logk12 logbn2

. 3.15 Substituting this into3.14,3.13holds by3.12.

The following example shows thatLemma 3.7fails if3.12does not hold.

Example 3.8. Let φ0 1 andφn 2φn−1 forn ≥ 1.Define a sequence{bn, n ≥ 1}bybn

φk1ifφk≤n < φk1.Then 0< bn ↑ ∞andbn≥nfor alln≥1.Sincebφnφn1, we obtain that

logbφn

logφn

logφn1

logφn

φnlog 2

logφn −→ ∞ 3.16

asn → ∞.Hence3.12does not hold. We also obtain that forn≥2, b2_φ_n

log2φn

∞

iφn logi

ib2_i ≥

b_φ2_n

log2φn

φn1−1

iφn logi

ib2_i

1

log2φn

φn1−1

iφn

logi i

≥ 1

log2φn

φn1

φn

logx x dx

1

2

_φ_n_{log 2} logφn

2 − 1

2 −→ ∞

3.17

asn → ∞.So3.13does not hold.

If{bn}satisfies2.1and3.12, then we can obtain the following SLLN.

{bn, n ≥ 1}be a nondecreasing unbounded sequence of positive numbers satisfying 2.1and3.12.

Suppose that the following conditions hold:

i∞_n₁logn2EX2

n/b2n<∞.

ii∞_n₁logn2ρn/n <∞.

Then1.3holds.

Proof. ByLemma 3.7andi, we get

∞

n1 EX_n2

∞

in1

logi

ib2_i ≤O1

∞

n1

log2n1EX2

n

b_n2₁ <∞, 3.18

(9)

Remark 3.10. Under condition 3.12,Theorem 3.9improves Theorem 3.1, since conditionii ofTheorem 3.9is weaker than conditioniiofTheorem 3.1.

If bn n for all n ≥ 1, then {bn} satisfies 2.1 and 3.12. Hence we can obtain the

following.

Corollary 3.11. Let{Xn, n≥ 1}be a sequence of square integrable random variables satisfying1.1.

Suppose that the following conditions hold.

i∞_n₁logn2EXn2/n2<∞.

ii∞_n₁logn2ρn/n <∞.

Then the SLLN holds. Namely,

n

i1Xi

n −→0a.s. 3.19 Remark 3.12. Lyons3proved an SLLN3.19under the conditions thatEX2nO1and

∞

n1 ρn

n <∞. 3.20

When EXn2 O1,condition iofCorollary 3.11 is obviously satisfied. Hu et al.1proved

an SLLN3.19under conditions3.21and3.22:

∞

n1

Hnϕ1

n2 <∞, 3.21

∞

n1 ρn

nϕ−1 <∞, 3.22

whereϕ 1√5/21.618· · ·is the golden ratio, andHx>0 is a nondecreasing function on0,∞such that EXn2 ≤ Hnfor alln ≥ 1.Conditioniiof Corollary 3.11is weaker than

3.22. In general, conditioniofCorollary 3.11is not comparable with3.21.

Acknowledgments

The author would like to thank the referees for helpful comments and suggestions that considerably improved the presentation of this paper. This work was supported by Korea

Science and Engineering Foundation KOSEF grant funded by Korea government MOST

no. R01-2007-000-20053-0.

References

1T.-C. Hu, A. Rosalsky, and A. I. Volodin, “On the golden ratio, strong law, and first passage problem,” The Mathematical Scientist, vol. 30, no. 2, pp. 77–86, 2005.

2T.-C. Hu, A. Rosalsky, and A. Volodin, “On convergence properties of sums of dependent random variables under second moment and covariance restrictions,”Statistics & Probability Letters. In press. 3R. Lyons, “Strong laws of large numbers for weakly correlated random variables,” Michigan

(10)

4T. K. Chandra, “Extensions of Rajchman’s strong law of large numbers,”Sankhy¯a. Series A, vol. 53, no. 1, pp. 118–121, 1991.

5F. M ´oricz, “The strong laws of large numbers for quasi-stationary sequences,” Zeitschrift f ¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 38, no. 3, pp. 223–236, 1977.

6F. M ´oricz, “SLLN and convergence rates for nearly orthogonal sequences of random variables,” Proceedings of the American Mathematical Society, vol. 95, no. 2, pp. 287–294, 1985.

7R. J. Serfling, “Convergence properties ofSnunder moment restrictions,”The Annals of Mathematical

Statistics, vol. 41, no. 4, pp. 1235–1248, 1970.

8R. J. Serfling, “On the strong law of large numbers and related results for quasistationary sequences,” Theory of Probability and Its Applications, vol. 25, no. 1, pp. 187–191, 1980.

9R. J. Serfling, “Moment inequalities for the maximum cumulative sum,”The Annals of Mathematical Statistics, vol. 41, no. 4, pp. 1227–1234, 1970.