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Volume 2008, Article ID 385362, 11 pagesdoi:10.1155/2008/385362 Research Article Exponential Inequalities for Positively Associated Random Variables and Applications Guodong Xing, 1 Shan

Volume 2008, Article ID 385362, 11 pages

doi:10.1155/2008/385362

Research Article

Exponential Inequalities for Positively Associated Random Variables and Applications

Guodong Xing, 1 Shanchao Yang, 2 and Ailin Liu 3

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

425100 Hunan, China

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

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

425100 Hunan, China

Correspondence should be addressed to Guodong Xing, xingguod@163.com

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 rate n −1/2 log log n 1/2 log n2

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 rate n −1/3 log n 5/3derived by Oliveira 2005 for the above case.

Copyright q 2008 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{X i , 1 ≤ i ≤ n} is said to be positively associated PA if

for every pair of disjoint subsets A1and A2of{1, 2, , n},

f1



X i , i ∈ A1



, f2



X j , j ∈ A2



whenever f1and f2are coordinatewise increasing and the covariance exists An infinite family

is positively associated if every finite subfamily is positively associated

i1 X i − EX i play a very important role in various proofs of limit theorems For positively associated

2 generalized later the previous results Recently, Ioannides and Roussas 3 established

a Bernstein-Hoeffding-type inequality for stationary and positively associated random

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

of n

i1 X i − EX i /n → 0 a.s is n −1/3 log n 5/3 under the rate of covariances supposed to be

geometrically decreasing, that is, ρ n for some 0 < ρ < 1 The convergence rate is partially

motivate us to establish some new exponential inequalities in order to improve the

rate n −1/2 log log n 1/2 log n2if 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

4 without the boundedness assumption

Throughout this paper, we always suppose that C denotes a positive constant which only

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 constant M Then for any λ > 0,





E



exp



λ

n

i1

X i −n

i1

E exp

λX i





1≤i<j≤n

X i , X j



Lemma 2.2 see 7 Let {Xi , i ≥ 1} be a positively associated sequence with zero mean and

∞

i1

v 1/2

2i

where vn  sup i ≥ 1

j:j−i ≥ nCov1/2 X i , X j  Then there exists a positive constant C such that

E max

1≤j≤n







j

i1

X i







2

≤ Cn

sup

i ≥ 1

EX i2 sup

i ≥ 1

EX i2

1/2

Remark 2.3 see condition 2.2 is quite weak In fact, it is satisfied only if vn ≤

Clog n−2log log n −2−ξ for some ξ > 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

and un  sup i ≥ 1

j:j−i ≥ nCovXi , X j  Also, for convenience, we define X ni by X ni  X i for

1≤ i ≤ n and X ni  0 for i > n, and let

X 1,i,n −c n

2I −∞,−c n /2



X ni



 X ni I −c n /2,c n /2



X ni



c n

2I c n /2,∞



X ni



X 2,i,n X ni−c n

2



I c n /2,∞



X ni



, X 3,i,n X ni c n

2



I −∞,−c n /2



X ni



for each n, i ≥ 1, where I A represents the characteristic function of the set A Consider now a sequence of natural numbers p n such that for each n ≥ 1, p n < n/2, and set r n  n/2p n  1 Define, then,

Y q,j,n  2j−1p n p n

i2j−1p n1



X q,i,n − EX q,i,n

, Z q,j,n 2jp n

i2j−1p n p n1



X q,i,n − EX q,i,n, 2.6

for q  1, 2, 3, j  1, 2, , r n, and

S q,n,od r n

j1

Y q,j,n , S q,n,ev  r n

j1

Clearly, n ≤ 2r n p n < 2n.

The proofs given later will be divided into the control of the bounded terms that

corre-spond to the index q  1 and the control of the unbounded terms, correcorre-sponding to the indices

q  2, 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 {X i , i ≥ 1} be a positively associated sequence Then on account of definitions 2.5,

2.6, 2.7, and for every λ > 0,





E

 exp

λS 1,n,od



−r n

j1

E exp

λY 1,j,n





 ≤ λ2nu



p n

 exp

λnc n



,





E

 exp

λS 1,n,ev



−r n

j1

E exp

λZ 1,j,n





 ≤ λ2nu



p n

 exp

λnc n



.

3.1

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

Lemma 3.2 Let {X i , i ≥ 1} be a positively associated sequence and let 2.2 hold If 0 < λpn c n ≤ 1

for λ > 0, then

r n

j1

E exp

λY 1,j,n



≤ expC1λ2nc2n

r n

j1

E exp

λZ 1,j,n



≤ expC1λ2nc2n

where C1is a constant, not depending on n.

Proof Since EY 1,j,n  0 and 0 < λp n c n≤ 1, we may have

E

exp

λY 1,j,n



 ∞

k0

E

λY 1,j,n

k

k2

E

λY 1,j,n

k

k!

≤ 1  EλY 1,j,n

2 ∞

k2

1

k! ≤ 1  λ2

EY 1,j,n2 ≤ expλ2EY 1,j,n2 

.

3.4

By this,Lemma 2.2and|X 1,i,n | ≤ c n /2,

r n

j1

E

exp

λY 1,j,n



≤ exp



λ2

r n

j1

EY 1,j,n2

≤ exp



Cλ2p n

r n

j1

sup

i ≥ 1

Var

X 1,i,n



i ≥ 1

Var

X 1,i,n

1/2

≤ exp



Cλ2p n

r n

j1

sup

i ≥ 1

EX21,i,n sup

i ≥ 1

EX 1,i,n2

1/2

≤ expCλ2r n p n



c n /22

≤ expC1λ2nc2n

3.5

as desired The proof is completed

Remark 3.3 The upper bound of 4, Lemma 3.1 is expλ2np n c2

the condition 0 < λp n c n ≤ 1, which is equivalent to 0 < λ ≤ 1/p n c n and enables us to get the

Lemma 3.4 Let {X i , i ≥ 1} be a positively associated sequence and let 2.2 hold If 0 < λpn c n ≤ 1

for λ > 0, then for any ε > 0,

P





n

i1



X 1,i,n − EX 1,i,n



 > nε ≤ 4



λ2nu

p n



e λnc n  e C1λ2nc2

n

where X 1,i,n and C1are just as in2.5 and 3.2

By Lemma 3.4 , one can show a result as follows.

Theorem 3.5 Let {X i , i ≥ 1} be a positively associated sequence and let 2.2 hold Suppose that

p n ≤ n/α log n for some α > 0, p n → ∞, and

log n

n 2α/3 p n c n2

exp αn log n

p n

1/2

u

p n



≤ C0< ∞, 3.7

where C0 is a constant which does not depend on n Set ε n  10/3αp n c2

n log n/n 1/2 Then there exists a positive constant C2, which only depends on α > 0, such that

P





n

i1



X 1,i,n − EX 1,i,n





Proof Let λ  10α log n/3nε n  α log n/np n c2

n1/2 and ε  ε ninLemma 3.4 Then it is obvious

that λp n c n ≤ 1 from p n ≤ n/α log n and that

e C1λ2nc2

n  exp C1α log n

p n



3α log n



λ2nu

p n



e λnc n  α log n

p n c2n

exp αn log n

p n

1/2

u

p n



≤ C2n 2α/3  C2exp 2

3α log n

Remark 3.6. 1 Let us compareTheorem 3.5with4, Theorem 3.6 Our result drops the strict stationarity of the positively associated random variables; and to obtain3.8, Oliveira 4 used the following condition:

log n

p n c2

n

exp αn log n

p n

1/2

u

p n



≤ C0< ∞. 3.12

Obviously,3.7 is weaker than 3.12

i1 X i − EX i /n than the

one of convergence in4 This is because εn  10/3αp n c n2log n/n 1/2, preventing us from

getting the convergence rate n −1/2 log log n 1/2 log n2 for the case of geometrically decreas-ing covariances So to obtain the above rate, we show another exponential inequality3.20 in

which ε n  p n c n



log log n log n/2n, permitting us to get the desired rate when we use

condi-tion3.19 instead of condition 3.7, which is weaker than condition 3.19 for the case α > 2/3,

0 < δ < 1/2, and p n ≤ 4  3δ2

n/α2log n log log n.

Now, let us consider3.8 again By Borel-Cantelli lemma, we need∞n1 e −α log n < ∞ for

some α > 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 {X i , i ≥ 1} be a positively associated sequence and let 2.2 hold Assume that

{ε n : n ≥ 1} is a positive real sequence which satisfies

p n c n log n

nε2

n

and for some  > 0 and δ > 0,

n −12δ log n

ε n

2 exp 21  3δcn log n

ε n 



u

p n

≤ C0< ∞. 3.14

Then there exists a positive constant C, which depends on  > 0 and δ > 0, such that

P





n

i1



X 1,i,n − EX 1,i,n





Proof Let λ  21  3δ log n/nε n  and let ε  ε n  inLemma 3.4 Then it is obvious that λpn c n≤

1 from3.13 and that

Also, we can get that

e C1λ2nc2

n  exp C141  3δ

2

c2

n log n

2nε2

n



by3.13, and that

λ2nu

p n



e λnc n  21  3δ



ε n

2

n−1exp 21  3δcn log n

ε n 



u

p n



≤ Cn 2δ  C exp2δ log n

3.18

Taking ε n  p n c n



result

Corollary 3.8 Let {X i , i ≥ 1} be a positively associated sequence and let 2.2 hold Suppose that pn

satisfies

n/log n ≤ p n < n/2 and for some  > 0 and δ > 0,

n1−2δ

p2

n c2

n log log nexp

⎛

p n



log log n

⎞

⎟

⎠ up n



≤ C0< ∞. 3.19

Then there exists a positive constant C3, which depends on  > 0 and δ > 0, such that

P





n

i1



X 1,i,n − EX 1,i,n





 > p n c n



log log n log n ≤ C3exp

4 Control of the unbounded terms

In this section, we will try ourselves to control the unbounded terms Firstly, it is obvious that

the variables X 2,i,n and X 3,i,n are positively associated but not bounded, even for fixed n This

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 of X 2,i,n or X 3,i,n The results we get are listed below

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

sup

i ≥ 1, |t|≤ω

E

e tX i

for some ω > 0 and let 2.2 hold Then for 0 < t ≤ ω,

P

 max 1≤j≤n







j

i1



X q,i,n − EX q,i,n



 > nε ≤ C



2M ω e −tc n /2

ntε2 , q  2, 3. 4.2

Proof Firstly, let us estimate EX q,i,n2 Without loss of generality, set q  2 We will assume Fx 

P X i > x Then by Markov’s inequality and sup i ≥ 1, |t|≤ω Ee tX i  ≤ M ω < ∞ for some ω > 0, it

follows that, for 0 < t ≤ ω,

Fx ≤ e −tx E

e tX i

Writing the mathematical expectation as a Stieltjes integral and integrating by parts, we have

EX22,i,n −



c n /2,∞

x − c n

2

2

dFx

 − x − c n

2

2

Fx

∞

c n /2





c n /2,∞

2 x − c n

2



Fxdx

 − lim

x→∞ x − c n

2

2

Fx 



c n /2,∞

2 x − c n

2



Fxdx





c n /2,∞

2 x − c n

2



Fxdx



c n /2,∞

x − c n

2



e −tx dx

 2M ω e −tc n /2

t2

4.4

by the inequality stated earlier Hence using4.4 andLemma 2.2, we have, for n large enough,

P



max

1≤j≤n







j

i1



X 2,i,n − EX 2,i,n



 > nε ≤

E max1≤j≤nj

i1



X 2,i,n − EX 2,i,n2

n2ε2

 supi ≥ 1Var

X 2,i,n



supi ≥ 1Var

X 2,i,n

1/2

n2ε2

 supi ≥ 1 EX2

2,i,nsupi ≥ 1 EX2

2,i,n

1/2

nε2



2M ω e −tc n /2

ntε2

4.5

This completes the proof of the lemma

Remark 4.2 Let {X i , i ≥ 1} be a positively associated sequence and let 2.2 hold as mentioned

from the following aspects

i The assumption of the stationarity of {X i , i ≥ 1} is dropped.

ii The sum in 4.2 is

max 1≤j≤n







j

i1



X q,i,n − EX q,i,n



, not







n

i1



X q,i,n − EX q,i,n



iii The upper bound of the exponential inequality in 4, Lemma 4.1 is 2Mω ne −t c n /t2ε2,

2M ω e −t c n /nt2ε2 Obviously, C

2M ω e −t c n /nt2ε2 ≤

2M ω ne −t c n /t2ε2for sufficiently large n That is, the upper bound inLemma 4.1is much lower than that of4, Lemma 4.1

t and c n

Corollary 4.3 Let {X i , i ≥ 1} be a positively associated sequence that satisfies sup i ≥ 1, |t|≤ω Ee tX i ≤

M ω < ∞ for some ω > 0 and let 2.2 hold Then

P



max 1≤j≤n







j

i1



X q,i,n − EX q,i,n



 > nε ≤ C



provided t  2α and c n  2 log n, and

P



max

1≤j≤n







j

i1



X q,i,n − EX q,i,n



 > nε ≤ C



2αnε2 exp

provided t  2α and c n  21  δ/αlog n, where α and δ are as in 3.8 and 3.13.

5 Strong convergences and rates

This section summarizes the results stated earlier In addition, we give a convergence rate for

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

1

n 2α/3 p n log nexp

αn log n

p n

1/2

u

p n



≤ C0< ∞ 5.1

for some α > 0, n/α log n ≥ p n → ∞ and let 2.2 hold Suppose that ε n is as in Theorem 3.5 and there exists ω > α that satisfies sup i ≥ 1, |t|≤ω Ee tX i  ≤ M ω < ∞ Then for sufficiently large n,

P





n

i1



X i − EX i





200α2p nlog3n



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

ii {X i , i ≥ 1} satisfies 1/p n log n exp{αn log n/p n1/2 }up n  ≤ C0 < ∞ Clearly, it

implies5.1

200α2p n log3n ≤ 2M ω n2/9α3p nlog3n for sufficiently large n.

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

Theorem 5.3 Let {X i , i ≥ 1} be a positively associated sequence satisfying 3.19 for n/log n ≤

p n < n/2, some  > 0, and δ > 0 and let 2.2 hold Suppose that supi ≥ 1, |t|≤ω Ee tX i  ≤ M ω < ∞ for some ω > α Then for n large enough,

P







n

i1



X i − EX i



 > 3n n ≤ C3C



2αn2n

 exp

where  n  p n c n

log log n log n/n and c n  21  δ/αlog n.

Applying Theorem 5.3 , one may have immediately some strong laws of large numbers by taking

p n √n and p n  n/4, respectively.

Corollary 5.4 Let {X i , i ≥ 1} be a positively associated sequence which satisfies supi ≥ 1, |t|≤ω Ee tX i  ≤ M ω < ∞ for some ω > α Then

n

i1



X i − EX i





n log log n log2n

provided that

expα√nu

√n

n 2δlog2n log log n ≤ C < ∞ for some α > 0 , δ > 0, 5.5

and2.2 holds; and

n

i1



X i − EX i



n

log log n log2n

provided that

u

n/4

n12δlog2n log log n ≤ C < ∞ for some δ > 0, 5.7

and2.2 holds

Finally, one gives some applications of Corollary 5.4

(1) Suppose now Cov X i , X j   Cρ |i−j| for some 0 < ρ < 1 Then v√

n ∼Cρ√n/2 and u√

n ∼Cρ√n , so 2.2 is satisfied and

exp

α√

n

u

√n 

∼Cρe α√

by choosing α > 0 with 0 < ρe α < 1 This means that one requires only 0 < α < −log ρ, not α >

8/3 in [ 4 ] It is due to Lemma 4.1 By5.8, one knows that 5.5 holds Hence one gets finally that

n

i1 X i −EX i /n → 0, a.s., converges at the rate n −1/2 log log n 1/2log2n which closes to the optimal achievable convergence rate for independent random variables under the Hartman-Wintner law of the iterated logarithm However, Oliveira [ 4 ] only got n −1/3log5/3 n for the case mentioned above Clearly, the convergence rate is much lower than ours.

(2) If Cov X i , X j   C|j − i| −τ for some τ > 2, or CovX i , X j   C|j − i|−2log−η |j − i| for some

η > 8, then it is clear that 5.7 and 2.2 can be satisfied Therefore By 5.6, 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

and the key Science Foundation of Hunan University of Science and Engineering

References

1 T Birkel, “Moment bounds for associated sequences,” Annals of Probability, vol 16, no 3, pp 1184–1193,

1988.

2 Q.-M Shao and H Yu, “Weak convergence for weighted empirical processes of dependent sequences,”

Annals of Probability, vol 24, no 4, pp 2098–2127, 1996.

3 D A Ioannides and G G Roussas, “Exponential inequality for associated random variables,” Statistics

& Probability Letters, vol 42, no 4, pp 423–431, 1999.

Lemma 3.2 Let {X i , i ≥ 1} be a positively associated sequence and let 2.2... 2α and c n  21  δ/αlog n, where α and δ are as in 3.8 and 3.13.

5...