Maximal Inequalities for Dependent Random Variables and Applications

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 ofn

i1Xi/bi

andn i1Xi/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≥1EXkXkn≤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

X_i

b_i converges a.s. asn−→ ∞ 1.2

and the strong law of large numbersSLLN _n

i1X_i

bn −→0 a.s. 1.3

Note that if 0< bn↑ ∞,then1.2implies1.3by the Kronecker lemma.

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 O1andb_n n.Without condition VarXn O1,Hu et al.1obtained an SLLN, whereb_n 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≥1 and some constantD >0. 2.1 Then for alln≥1, m > n,andδ >0,

m−1

in

m ji1

EXiX_j

b_ib_j ≤ D²C_δ

max{log 2^δ,logn^δ}

m−n

k1

ρ_k

k1logk^1δ, 2.2

whereC_δ2^δ1max{1, δ^δe^−δ}.

Proof. For simplicity of notation, letI_{n,m m−1}

in_m

ji1EXiX_j/bib_j.Then we get by1.1 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

ρ_k k

m−k

in

1 i − 1

ik

≤D²

m−n

k1

ρ_k k

nk−1

in

1 i.

2.3

We next estimate_nk−1

in 1/i.Ifn1,then

nk−1

in

1 i ^k

i1

1 i ≤1

_k

1

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

1k n

≤

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩ log

1 1

√n

≤ logn^δ logn^δ√

n ≤2δ^δe^−δ

logn^δ ≤2δ^δe^−δ1logk^1δ

logn^δ , if 1≤k≤√ n,

log

1k n

2 logk^δ

logn^δ ≤2^δlogk^1δ

logn^δ ≤2^δ1logk^1δ

logn^δ , ifk >√ n.

2.6 Thus, we have the desired estimate forIn,m:

I_n,m≤

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩ D²

m−n

k1

ρk

k1logk^1δ, ifn1,

D²

m−n

k1

ρ_k k

2 max{2^δ,2δ^δe^−δ}

logn^δ 1logk^1δ, ifn≥2,

≤ D²2^δ1max{1, δ^δe^−δ} max{log 2^δ,logn^δ}

m−n

k1

ρk

k1logk^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

1≤k≤nmax

ak ia1

X_i

2

≤log 2n log 2

2_an

ia1

EX_i²2

an−1

ia1 an

ji1

EXiX_j

. 2.8

Proof. LetF_a,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

EXiXj. 2.9

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 _an

ia1

Xi

₂

≤gFa,n. 2.11

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

E

max1≤k≤n

ak

ia1

Xi

2

≤

log 2n log 2

2

gFa,n. 2.12

Thus, the result is proved.

Combining Lemmas 2.1 and2.2 gives 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

n≤i≤mmax

i jn

X_j bj

2

≤log2m−n1 log 2

2_m

in

EX²_i

b²_i 2D²C_δ

max{log 2^δ,logn^δ}

m−n

k1

ρk

k1logk^1δ

, 2.13 whereCδ2^δ1max{1, δ^δe^−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_∞

n1logn²EX_n²/b²_n<∞, ii_∞

n1logn^4δρ_n/n <∞for someδ >0.

Then1.2holds. Furthermore, if 0< bn↑ ∞,then1.3holds.

Proof. As noted in the introduction, if 0 < bn ↑ ∞,then1.2implies1.3. To prove1.2, let Sn_n

i1Xi/bi.ByLemma 2.1withδreplaced by 3δ, we have that form > n, ESm−Sn^{2 m}

in1

EX²_i b²_i 2

m−1

in1

m ji1

EXiXj

bibj

≤ ^m

in1

EX²_i

b²_i 2D²C3δ

max{log 2^3δ,logn^3δ}

m−n−1

k1

ρ_k

k1logk^4δ

≤ ^∞

in1

EX²_i

b²_i 2D²C3δ

max{log 2^3δ,logn^3δ} ∞ k1

ρ_k

k1logk^4δ−→0

3.1

asn → ∞byiandii. HereC3δ2^δ4max{1,3δ^3δe^−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

2ⁿmax<k≤2ⁿ¹|Sk−S₂ⁿ| −→0 a.s. asn−→ ∞. 3.2 UsingLemma 2.3,i, andii, we get that

∞ n1

P

2ⁿmax<k≤2ⁿ¹|Sk−S₂ⁿ|>

≤ 1 ²

∞ n1

E

2ⁿmax<k≤2ⁿ¹|Sk−S2ⁿ|²

≤ 1 ²

∞ n1

log 2ⁿ¹ log 2

2 _2n1

i2ⁿ1

EX_i²

b_i² 2D²C_3δ log2ⁿ1^3δ

2ⁿ−1 k1

ρk

k 1logk^4δ

≤ 1

²log 2² ∞

i3

log2i²EX²_i

b_i² 2D²C3δ

²log 2^3δ ∞ n1

n1² n^3δ

∞ k1

ρ_k

k 1logk^4δ<∞.

3.3

Then3.2follows by the Borel-Cantelli lemma.

Remark 3.2. Hu et al.2provedTheorem 3.1underiandii. ii _∞

n1ρ_n/n^q<∞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< b_n↑ ∞.

Theorem 3.3. 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. Suppose that the following conditions hold:

i_∞

n1logn²EX_n²/b²_n<∞,

ii_∞

n1ρ_n∞

in1logi²/b_i²<∞, iii_∞

n1EX²_n∞

in1logi/ib²_i<∞.

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

max1≤i≤n

i j1

Xj

r

≤ⁿ

i1

αi, for some constantr >0. 3.4 If_∞

n1α_n/b_n^r <∞,then1.3holds.

Proof ofTheorem 3.3. FromLemma 2.2,

E

max1≤i≤n

i j1

X_j

2

≤log 2n log 2

2 _n

i1

EX_i²2 n−1

i1

n ji1

EXiX_j

. 3.5

Defineαn log 2n/log 2²An−log 2n−1/log 2²An−1forn ≥1,whereA0 0 andAn _n

i1EX_i²2_n−1

i1_n

ji1EXiX_jforn≥1.ThenEmax1≤i≤n|_i

j1X_j|²≤_n

i1α_iand αn

log 2n log 2

₂

An−An−1 An−1

log 2n log 2

₂

−

log 2n−1 log 2

2

log 2n log 2

2

EX_n²2

n−1

i1

EXiX_n

log 2n log 2

₂

−

log 2n−1 log 2

2n−1 i1

EX_i²2

n−2

i1 n−1

ji1

EXiXj

.

3.6

ByLemma 3.4, it is enough to show that ∞ n1

log 2n²EX_n²

b²_n <∞, 3.7

∞ n1

log 2n² b²_n

n−1

i1

EXiXn<∞, 3.8

∞ n2

log 2n²−log 2n−1² b²_n

n−1

i1

EX²_i <∞, 3.9

∞ n3

log 2n²−log 2n−1² b²_n

n−2 i1

n−1 ji1

EXiXj<∞. 3.10 Clearly3.7holds by i. It is easy to see that3.8–3.10hold, and the detailed proofs are omitted.

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_∞

n1logn²EX_n²/b²_n<∞, ii_∞

n1logn²ρ_n/n <∞, iii_∞

n1EX²_n∞

in1logi/ib²_i<∞.

Then1.3holds.

Proof. By2.1, we have that ∞ n1

ρn

∞ in1

logi² b²_i ≤D²

∞ n1

ρn

∞ in1

logi² i² ≤C

∞ n1

ρnlogn²

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 ofTheorem 3.3 if conditions 2.1and 3.12on{bn} are satisfied. IfC1n ≤ bn ≤ C2n^α 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≤b_ifor alli≥1.

Let fixn.For eachk ≥1,definem_kbym_kmin{i ≥n:b_i ≥kb_n}.Thenb_mk ≥kb_nand nm1≤m2≤ · · ·.It follows that

b²_n log²n

∞ in

logi ib²_i b²_n

log²n ∞ k1

mk1−1 imk

logi ib²_i ≤ b²_n

log²n ∞ k1

1 b²_mk

mk1−1 imk

logi

i by 0< bn↑

≤ b²_n log²n

∞ k1

1 kbn²

mk1−1 imk

logi i

≤ 1 log²n

∞ k1

1 k²

₃

i1

logi

i

_mk1−1

3

logx x dx

≤ 1 log²n

∞ k1

1 k²

log 2 2 log 3

3 logmk1−1² 2

,

3.14

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 logb_n²

. 3.15

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

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

Example 3.8. Letφ0 1 andφn 2^φn−1forn ≥ 1.Define a sequence{bn, n ≥ 1}bybn φk1ifφk≤n < φk1.Then 0< b_n↑ ∞andb_n≥nfor alln≥1.Sinceb_φnφn1, we obtain that

logbφn

logφn logφn1

logφn φnlog 2

logφn −→ ∞ 3.16

asn → ∞.Hence3.12does not hold. We also obtain that forn≥2, b_φn²

log²φn ∞ iφn

logi

ib²_i ≥ b²_φn log²φn

φn1−1

iφn

logi ib_i² 1

log²φn

φn1−1

iφn

logi i

≥ 1

log²φn _φn1

φn

logx x dx 1

2

φnlog 2 logφn

2

−1 2 −→ ∞

3.17

asn → ∞.So3.13does not hold.

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

Theorem 3.9. 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 satisfying2.1and3.12.

Suppose that the following conditions hold:

i_∞

n1logn²EX_n²/b²_n<∞.

ii_∞

n1logn²ρn/n <∞.

Then1.3holds.

Proof. ByLemma 3.7andi, we get ∞

n1

EX²_n ∞ in1

logi

ib²_i≤O1^∞

n1

log²n1EX_n²

b²_n1 <∞, 3.18

since logn1/bn1≤logn1/bn≤2 logn/b_nifn≥2.The result follows byCorollary 3.5.

Remark 3.10. Under condition3.12,Theorem 3.9improvesTheorem 3.1, since conditionii ofTheorem 3.9is weaker than conditioniiofTheorem 3.1.

If bn nfor all n ≥ 1,then {bn} satisfies2.1and 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_∞

n1logn²EX_n²/n²<∞.

ii_∞

n1logn²ρ_n/n <∞.

Then the SLLN holds. Namely,

_n

i1Xi

n −→0 a.s. 3.19

Remark 3.12. Lyons3proved an SLLN3.19under the conditions thatEX²_nO1and ∞

n1

ρn

n <∞. 3.20

WhenEX_n² O1,conditioniofCorollary 3.11is obviously satisfied. Hu et al.1proved an SLLN3.19under conditions3.21and3.22:

∞ n1

Hn^ϕ1

n² <∞, 3.21

∞ n1

ρn

n^ϕ−1 <∞, 3.22

whereϕ 1√

5/21.618· · ·is the golden ratio, andHx>0 is a nondecreasing function on0,∞such thatEX_n² ≤Hnfor alln ≥1.ConditioniiofCorollary 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 FoundationKOSEFgrant funded by Korea governmentMOST no. R01-2007-000-20053-0.

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