Exponential Inequalities for Positively Associated Random Variables and Applications

Volume 2008, Article ID 385362,11pages doi:10.1155/2008/385362

Research Article

Exponential Inequalities for Positively Associated Random Variables and Applications

Guodong Xing,¹Shanchao Yang,²and Ailin Liu³

1Department of Mathematics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China

2Department of Mathematics, Guangxi Normal University, Guilin, 541004 Guangxi, China

3Department of Physics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China

Correspondence should be addressed to Guodong Xing,[email protected] Received 1 January 2008; Accepted 6 March 2008

Recommended by Jewgeni Dshalalow

We establish some exponential inequalities for positively associated random variables without the boundedness assumption. These inequalities improve the corresponding results obtained by Oliveira2005. By one of the inequalities, we obtain the convergence raten^−1/2log logn^1/2logn² for the case of geometrically decreasing covariances, which closes to the optimal achievable conver- gence rate for independent random variables under the Hartman-Wintner law of the iterated log- arithm and improves the convergence raten^−1/3logn^5/3derived by Oliveira2005for the above case.

Copyrightq2008 Guodong Xing et al. 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

A finite family of random variables{Xi, 1 ≤ i ≤n}is said to be positively associatedPAif for every pair of disjoint subsetsA1andA2of{1,2, . . . , n},

Cov f1

Xi, i∈A1

, f2

Xj, j∈A2

≥ 0 1.1

wheneverf1andf2are coordinatewise increasing and the covariance exists. An infinite family is positively associated if every finite subfamily is positively associated.

The exponential inequalities and moment inequalities for partial sum_n

i1Xi−EXi play a very important role in various proofs of limit theorems. For positively associated random variables, Birkel1seems the first to get some moment inequalities. Shao and Yu 2 generalized later the previous results. Recently, Ioannides and Roussas 3 established a Bernstein-Hoeffding-type inequality for stationary and positively associated random vari- ables being bounded; and Oliveira4gave a similar inequality dropping the boundedness

assumption by the existence of Laplace transforms. By the inequality, he obtained that the rate of_n

i1Xi−EXi/n → 0a.s.isn^−1/3logn^5/3 under the rate of covariances supposed to be geometrically decreasing, that is,ρⁿfor some 0< ρ < 1. The convergence rate is partially im- proved by Yang and Chen5only for positively associated random variables being bounded.

Furthermore, the rate of convergence in4is even lower than that obtained by3. These motivate us to establish some new exponential inequalities in order to improve the inequali- ties and the convergence rate which4obtained without the boundedness assumption. It is the main purpose of this paper. Our inequalities in Sections3–5 improve the corresponding results in 4. Moreover, by Corollary 5.4 which can be seen in Section 5, we may get the raten^−1/2log logn^1/2logn²if the rate of covariances is geometrically decreasing. The result closes to the optimal achievable convergence rate for independent random variables under the Hartman-Wintner law of the iterated logarithm and improves the relevant result obtained by 4without the boundedness assumption.

Throughout this paper, we always suppose thatCdenotes a positive constant which only depends on some given numbers,xdenotes the integral of x; and this paper is organized as follows. Section 2 contains some lemmas used later in the proof of theorems, and some notations.Section 3 studies the truncated part giving conditions on the truncating sequence to enable the proof of some exponential inequalities for these terms.Section 4treats the tails left aside from the truncation.Section 5summarizes the partial results into some theorems and gives some applications.

2. Some lemmas and notations Firstly, we quote two lemmas as follows.

Lemma 2.1 see 6. Let {Xi,1 ≤ i ≤ n}be positively associated random variables bounded by a constantM. Then for anyλ >0,

E

exp

λ

n i1

Xi

−ⁿ

i1

E exp

λXi

≤λ²expnλM

1≤i<j≤n

Cov Xi, Xj

. 2.1

Lemma 2.2see7. Let{Xi, i ≥ 1}be a positively associated sequence with zero mean and

∞ i1

v^1/2 2ⁱ

<∞, 2.2

wherevn sup_i≥1

j:j−i≥nCov^1/2Xi, Xj. Then there exists a positive constantCsuch that

Emax

1≤j≤n

j i1

Xi

2

≤Cn

sup

i≥1

EX_i² sup

i≥1

EX_i² 1/2

. 2.3

Remark 2.3 see condition 2.2 is quite weak. In fact, it is satisfied only if vn ≤ Clogn⁻²log logn^−2−ξ for someξ > 0. So it is weaker than the corresponding condition in 1,2.

For the formulation of the assumptions to be made in this paper, some notations are required. Thus letcn, n ≥ 1 be a sequence of nonnegative real numbers such thatcn → ∞ andun sup_i≥1

j:j−i≥nCovXi, Xj. Also, for convenience, we defineXni byXni Xi for 1≤i≤nandXni0 fori > n, and let

X1,i,n−cn

2I−∞,−cn/2 Xni

XniI−cn/2,cn/2 Xni

cn

2Icn/2,∞

Xni

, 2.4

X2,i,n

Xni−cn

2

Icn/2,∞

Xni

, X3,i,n

Xnicn

2

I−∞,−cn/2 Xni

, 2.5

for eachn, i ≥ 1, whereIArepresents the characteristic function of the setA. Consider now a sequence of natural numberspnsuch that for eachn ≥ 1, pn< n/2, and setrn n/2pn 1.

Define, then,

Yq,j,n

2j−1pnpn

i2j−1pn1

Xq,i,n−E Xq,i,n

, Zq,j,n

2jpn

i2j−1pnpn1

Xq,i,n−EXq,i,n

, 2.6

forq1,2,3,j1,2, . . . , rn, and

Sq,n,od ^rn

j1

Yq,j,n, Sq,n,ev ^rn

j1

Zq,j,n. 2.7

Clearly,n≤2rnpn<2n.

The proofs given later will be divided into the control of the bounded terms that corre- spond to the indexq1 and the control of the unbounded terms, corresponding to the indices q2,3.

3. Control of the bounded terms

In this section, we will work hard to control the bounded terms. For this purpose, we give some lemmas as follows.

Lemma 3.1. Let{Xi, i ≥ 1}be a positively associated sequence. Then on account of definitions2.5, 2.6,2.7, and for everyλ >0,

E

exp λS1,n,od

−^rn

j1E exp

λY1,j,n

≤λ²nu pn

exp λncn

,

E

exp λS1,n,ev

−^rn

j1

E exp

λZ1,j,n

≤λ²nu pn

exp λncn

.

3.1

Proof. Similarly to the proof of Lemma 3.2 in4, it is omitted here.

Lemma 3.2. Let{Xi, i ≥ 1}be a positively associated sequence and let2.2hold. If 0< λpncn≤1 forλ >0, then

rn

j1

E exp

λY1,j,n

≤exp

C1λ²nc²_n

, 3.2

rn

j1E exp

λZ1,j,n

≤exp

C1λ²nc²n

, 3.3

whereC1is a constant, not depending onn.

Proof. SinceEY1,j,n0 and 0< λpncn≤1, we may have

E exp

λY1,j,n ^∞

k0

E λY1,j,n

_k

k! 1 ^∞

k2

E λY1,j,n

_k k!

≤1E λY1,j,n

₂^∞

k2

1

k!≤1λ²EY_1,j,n² ≤exp

λ²EY_1,j,n² .

3.4

By this,Lemma 2.2and|X1,i,n| ≤cn/2,

rn

j1

E exp

λY1,j,n

≤exp

λ²

rn

j1

EY_1,j,n²

≤exp

Cλ²pn rn

j1

sup

i≥1

Var X1,i,n

sup

i≥1

Var X1,i,n

^1/2

≤exp

Cλ²pn rn

j1

sup

i≥1

EX_1,i,n²

sup

i≥1 EX_1,i,n²

_1/2

≤exp

Cλ²rnpn cn/22

≤exp

C1λ²nc²_n

3.5

as desired. The proof is completed.

Remark 3.3. The upper bound of4, Lemma 3.1is expλ²npnc²_n, and so the upper bound of Lemma 3.1is much sharper than that of4whenpn → ∞, this is the reason why we choose the condition 0< λpncn ≤1, which is equivalent to 0< λ≤1/pncnand enables us to get the desired upper bound byLemma 2.2.

Combining Lemmas3.1and3.2yields easily the following result.

Lemma 3.4. Let{Xi, i ≥ 1}be a positively associated sequence and let2.2hold. If 0< λpncn≤1 forλ >0, then for anyε >0,

P

n i1

X1,i,n−EX1,i,n > nε

≤4 λ²nu

pn

e^λncne^C1λ2nc2n

e^−nλε/2, 3.6 whereX1,i,nandC1are just as in2.5and3.2.

ByLemma 3.4, one can show a result as follows.

Theorem 3.5. Let{Xi, i ≥ 1}be a positively associated sequence and let2.2hold. Suppose that pn≤n/αlognfor someα >0,pn→ ∞, and

logn n^2α/3pnc²n

exp

αnlogn pn

1/2 u

pn

≤C0<∞, 3.7

whereC0 is a constant which does not depend onn. Setεn 10/3αpnc²_nlogn/n^1/2. Then there exists a positive constantC2, which only depends onα >0, such that

P

n i1

X1,i,n−EX1,i,n

> nεn

≤C2exp−αlogn. 3.8

Proof. Letλ10αlogn/3nεn αlogn/npnc²_n^1/2andεεninLemma 3.4. Then it is obvious thatλpncn≤1 frompn≤n/αlognand that

e^−nλεn/2e^−5/3αlogn. 3.9

Noting thatpn→ ∞, we may have

e^C1λ2nc2n exp

C1αlogn pn

≤exp 2

3αlogn

, 3.10

λ²nu pn

e^λncn αlogn pncn²

exp

αnlogn pn

1/2 u

pn

≤C2n^2α/3C2exp 2

3αlogn

3.11

by3.7. Combining3.9–3.11, we can get3.8byLemma 3.4. The proof is completed.

Remark 3.6. 1Let us compareTheorem 3.5with4, Theorem 3.6. Our result drops the strict stationarity of the positively associated random variables; and to obtain3.8, Oliveira4used the following condition:

logn pncn²

exp

αnlogn pn

1/2 u

pn

≤C0<∞. 3.12

Obviously,3.7is weaker than3.12.

2AlthoughTheorem 3.5holds under weaker conditions, it cannot make us get a much faster convergence rate for the almost sure convergence to zero of_n

i1Xi−EXi/nthan the one of convergence in4. This is becauseεn 10/3αpnc²_nlogn/n^1/2, preventing us from getting the convergence raten^−1/2log logn^1/2logn² for the case of geometrically decreas- ing covariances. So to obtain the above rate, we show another exponential inequality3.20in whichεnpncn

log lognlogn/2n, permitting us to get the desired rate when we use condi- tion3.19instead of condition3.7, which is weaker than condition3.19for the caseα >2/3, 0< δ <1/2, andpn≤43δ²n/α²lognlog logn.

Now, let us consider3.8again. By Borel-Cantelli lemma, we need_∞

n1e^−αlogn<∞for someα >0 in order to get strong law of large numbers. However, it is not true for 0< α≤1. To avoid this case, we show another exponential inequality.

Theorem 3.7. Let{Xi, i ≥ 1}be a positively associated sequence and let2.2hold. Assume that {εn:n ≥ 1}is a positive real sequence which satisfies

pncnlogn

nεn −→0, c²_nlogn nε²n

−→0, 3.13

and for some >0 andδ >0,

n^−12δ logn

εn

2

exp

213δcnlogn εn

u

pn

≤C0<∞. 3.14

Then there exists a positive constantC, which depends on >0 andδ >0, such that

P

n i1

X1,i,n−EX1,i,n

> nεn

≤Cexp

−1δlogn

. 3.15

Proof. Letλ213δlogn/nεnand letεεninLemma 3.4. Then it is obvious thatλpncn≤ 1 from3.13and that

e^−nλε/2e^−nλεn/2e^−13δlogn. 3.16

Also, we can get that

e^C1λ2nc2n exp

C1413δ²c²_nlogn ²nε²n

≤exp2δlogn 3.17

by3.13, and that

λ²nu pn

e^λncn213δ

2logn εn

2

n⁻¹exp

213δcnlogn εn

u

pn

≤Cn^2δCexp2δlogn

3.18

by3.14. Combining3.16–3.18, we can obtain3.15byLemma 3.4.

Taking εn pncn

log lognlogn/2nin Theorem 3.7, we can get easily the following result.

Corollary 3.8. Let{Xi, i ≥ 1}be a positively associated sequence and let2.2hold. Suppose thatpn

satisfies

n/logn≤pn< n/2 and for some >0 andδ >0,

n^1−2δ

p²nc²nlog lognexp

⎛

⎜⎝ 413δn pn

log logn

⎞

⎟⎠u pn

≤C0<∞. 3.19

Then there exists a positive constantC3, which depends on >0 andδ >0, such that

P

n i1

X1,i,n−EX1,i,n

> pncn

log lognlogn

≤C3exp

−1δlogn

. 3.20

4. Control of the unbounded terms

In this section, we will try ourselves to control the unbounded terms. Firstly, it is obvious that the variablesX2,i,nandX3,i,nare positively associated but not bounded, even for fixedn. This means thatLemma 3.1cannot be applied to the sum of such terms. While we may note that these variables depend only on the tails of distribution of the original variables. Hence by controlling the decrease rate of these tails, we may give some exponential inequalities for the sums ofX2,i,norX3,i,n. The results we get are listed below.

Lemma 4.1. Let{Xi, i ≥ 1}be a positively associated sequence that satisfies sup

i≥1,|t|≤ωE e^tXi

≤Mω<∞ 4.1

for someω >0 and let2.2hold. Then for 0< t≤ω, P

max1≤j≤n

j i1

Xq,i,n−EXq,i,n

> nε

≤C

2Mωe^−tcn/2

ntε² , q2,3. 4.2

Proof. Firstly, let us estimateEX_q,i,n² . Without loss of generality, setq2. We will assumeFx PXi > x. Then by Markov’s inequality and sup_{i≥1,|t|≤ω}Ee^tXi≤Mω <∞for someω >0, it follows that, for 0< t≤ω,

Fx≤e^−txE e^tXi

≤Mωe^−tx. 4.3

Writing the mathematical expectation as a Stieltjes integral and integrating by parts, we have EX_2,i,n² −

cn/2,∞

x−cn

2 2

dFx −

x−cn

2 2

Fx

^∞_c

n/2

cn/2,∞2

x−cn

2

Fxdx −lim

x→∞

x−cn

2 ₂

Fx

cn/2,∞2

x−cn

2

Fxdx

cn/2,∞2

x−cn

2

Fxdx

≤2Mω

cn/2,∞

x−cn

2

e^−txdx

2Mωe^−tcn/2 t²

4.4

by the inequality stated earlier. Hence using4.4andLemma 2.2, we have, fornlarge enough,

P

max1≤j≤n

j i1

X2,i,n−EX2,i,n

> nε

≤ Emax1≤j≤n^j_i1

X2,i,n−EX2,i,n² n²ε²

≤ Cn

sup_i≥1Var X2,i,n

sup_i≥1Var X2,i,n

_1/2 n²ε²

≤ C

sup_i≥1EX_2,i,n²

sup_i≥1EX_2,i,n² 1/2 nε²

≤ C

2Mωe^−tcn/2 ntε²

4.5

This completes the proof of the lemma.

Remark 4.2. Let{Xi, i ≥ 1}be a positively associated sequence and let2.2holdas mentioned above, it is a quite weak condition. ThenLemma 4.1improves the corresponding result in4 from the following aspects.

iThe assumption of the stationarity of{Xi, i ≥ 1}is dropped.

iiThe sum in4.2is

max1≤j≤n

j i1

Xq,i,n−EXq,i,n , not

n i1

Xq,i,n−EXq,i,n

in4. 4.6 iiiThe upper bound of the exponential inequality in4, Lemma 4.1is 2Mωne^−tcn/t²ε², where cn → ∞. So, assuming cn 4cn in the inequality 4.2, we can obtain that the upper bound of our inequality is C

2Mωe^−tcn/nt²ε². Obviously, C

2Mωe^−tcn/nt²ε² ≤ 2Mωne^−tcn/t²ε²for sufficiently largen. That is, the upper bound inLemma 4.1is much lower than that of4, Lemma 4.1.

ApplyingLemma 4.1, one can get immediately the following result by taking values for tandcn.

Corollary 4.3. Let{Xi, i ≥ 1}be a positively associated sequence that satisfies sup_{i≥1,|t|≤ω}Ee^tXi≤ Mω<∞for someω >0 and let2.2hold. Then

P

max1≤j≤n

j i1

Xq,i,n−EXq,i,n > nε

≤ C 2Mω

2αnε² exp−αlogn, q2,3, 4.7 providedt2αandcn2 logn, and

P

max1≤j≤n

j i1

Xq,i,n−EXq,i,n

> nε

≤ C 2Mω

2αnε² exp

−1δlogn

, q2,3, 4.8

providedt2αandcn 21δ/αlogn, whereαandδare as in3.8and3.13.

5. Strong convergences and rates

This section summarizes the results stated earlier. In addition, we give a convergence rate for geometrically decreasing covariances, which improves the relevant one obtained by4.

Theorem 5.1. Let{Xi, i ≥ 1}be a positively associated sequence satisfying 1

n^2α/3pnlognexp

αnlogn pn

1/2 u

pn

≤C0<∞ 5.1

for someα >0,n/αlogn ≥ pn→ ∞and let2.2hold. Suppose thatεnis as inTheorem 3.5and there existsω > αthat satisfies sup_{i≥1,|t|≤ω}Ee^tXi≤Mω<∞. Then for sufficiently largen,

P

n i1

Xi−EXi

>3nεn

≤

C2 9C 2Mω

200α²pnlog³n

exp−αlogn. 5.2

Proof. CombiningTheorem 3.5andCorollary 4.3yields the desired result5.2.

Remark 5.2. Theorem 5.1improves4, Theorem 5.1, because the latter uses the following more restrictive conditions.

i{Xi, i ≥ 1}is a strictly stationary sequence.

ii{Xi, i ≥ 1} satisfies1/pnlognexp{αnlogn/pn^1/2}upn ≤ C0 < ∞. Clearly, it implies5.1.

iiiThe latter has a higher upper bound than our result, because 9C 2Mω/ 200α²pnlog³n≤2Mωn²/9α³pnlog³nfor sufficiently largen.

Combining Corollaries3.8and4.3, we may get easily the following result.

Theorem 5.3. Let{Xi, i ≥ 1}be a positively associated sequence satisfying3.19for

n/logn ≤ pn< n/2, some >0, andδ >0 and let2.2hold. Suppose that sup_{i≥1,|t|≤ω}Ee^tXi≤Mω<∞for someω > α. Then fornlarge enough,

P

n i1

Xi−EXi

>3nn

≤

C3C 2Mω

2αn²n

exp

−1δlogn

, 5.3

wherenpncn

log lognlogn/nandcn 21δ/αlogn.

ApplyingTheorem 5.3, one may have immediately some strong laws of large numbers by taking pn √

nandpn n/4, respectively.

Corollary 5.4. Let {Xi, i ≥ 1} be a positively associated sequence which satisfies sup_{i≥1,|t|≤ω}Ee^tXi≤Mω<∞for someω > α. Then

_n

i1

Xi−EXi

nlog lognlog²n

−→0, a.s., 5.4

provided that

expα√ nu

√ n

n^2δlog²nlog logn ≤C <∞ for someα >0 ,δ >0, 5.5 and2.2holds; and

_n

i1

Xi−EXi

n

log lognlog²n

−→0, a.s., 5.6

provided that

u n/4

n^12δlog²nlog logn≤C <∞ for someδ >0, 5.7 and2.2holds.

Finally, one gives some applications ofCorollary 5.4.

(1) Suppose now CovXi, Xj Cρ^|i−j| for some 0 < ρ < 1. Then v√

n∼Cρ^√n/2 and

u√

n∼Cρ^√n,so2.2is satisfied and exp

α√ n

u √

n

∼C

ρe^α√n−→0 5.8

by choosingα > 0 with 0 < ρe^α < 1. This means that one requires only 0 < α < −logρ, not α >

8/3 in [4]. It is due toLemma 4.1. By5.8, one knows that5.5holds. Hence one gets finally that _n

i1Xi−EXi/n→0, a.s.,converges at the raten^−1/2log logn^1/2log²nwhich closes to the optimal achievable convergence rate for independent random variables under the Hartman-Wintner law of the iterated logarithm. However, Oliveira [4] only gotn^−1/3log^5/3nfor the case mentioned above. Clearly, the convergence rate is much lower than ours.

(2) If CovXi, Xj C|j−i|^−τfor someτ >2, or CovXi, Xj C|j−i|⁻²log^−η|j−i|for some η >8, then it is clear that5.7and2.2can be satisfied. Therefore By5.6, one does have almost sure convergence but without rates. The explicit reason could be seen in [4].

Acknowledgments

The authors thank the referees for their careful reading and valuable comments that improved presentation of the manuscript. This work is supported by the National Science Foundation of ChinaGrant no. 10161004, the Natural Science Foundation of GuangxiGrant no. 0728091, and the key Science Foundation of Hunan University of Science and Engineering.

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