Maximal inequality for ψ -mixing sequences and its applications

Applied Mathematics Letters 23 (2010) 1156–1161

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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

Maximal inequality for ψ -mixing sequences and its applicationsI Wang Xuejun, Hu Shuhe ∗ , Shen Yan, Yang Wenzhi School of Mathematical Science, Anhui University, Hefei 230039, China

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Article history: Received 21 April 2009 Received in revised form 27 January 2010 Accepted 7 April 2010 Keywords: Maximal inequality Strong law of large numbers Growth rate Integrability of supremum ψ -mixing sequences

First, a maximal inequality for ψ -mixing sequences is given. By using the maximal inequality, we study the convergence properties. A Hájek-Rényi-type inequality, strong law of large numbers, strong growth rate and the integrability of the supremum for ψ -mixing sequences are obtained. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Let {Xn , n ≥ 1} be a sequence of random variables defined on a fixed probability space (Ω , F , P ) and Sn = each n ≥ 1. Let n and m be positive integers. Write Fnm = σ (Xi , n ≤ i ≤ m). Given σ -algebras B , R in F , let

ψ(B , R) = ϕ(B , R) =

|P (AB) − P (A)P (B)| , P (A)P (B) A∈B ,B∈R,P (A)P (B)>0 sup

sup

A∈B ,B∈R,P (A)>0

Pn

i =1

Xi for

(1.1)

|P (B|A) − P (B)|.

(1.2)

Define the mixing coefficients by

ψ(n) = sup ψ(F1k , Fk∞ +n ); k≥1

ϕ(n) = sup ϕ(F1k , Fk∞ +n ),

n ≥ 0.

k≥1

Definition 1.1. A sequence {Xn , n ≥ 1} of random variables is said to be a ψ -mixing (ϕ -mixing) sequence of random variables if ψ(n) ↓ 0 (ϕ(n) ↓ 0) as n → ∞. It is easily seen that ϕ(n) ≤ ψ(n). Therefore, the family of ϕ -mixing contains ψ -mixing as a special case. ψ -mixing random variables were introduced by Blum, et al. [1] and some applications have been found. See for example, Blum, et al. [1] for the strong law of large numbers, Yang [2] for almost sure convergence of weighted sums, etc. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired. The main purpose of this paper is to study the maximal inequality for ψ -mixing sequences, by which we can get a

I Supported by the National Natural Science Foundation of China (Grant No. 10871001, 60803059), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), Youth Science Research Fund of Anhui University (2009QN011A). ∗ Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (S. Hu), [email protected] (Y. Shen), [email protected] (W. Yang).

0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.04.010

X. Wang et al. / Applied Mathematics Letters 23 (2010) 1156–1161

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Hájek–Rényi-type inequality, strong law of large numbers, strong growth rate and the integrability of the supremum for ψ mixing sequences. For more details about Hájek–Rényi-type inequality, one can refer to Fazekaa and Klesov [3], Fazekas [4], Tómács and Líbor [5], and so on. The main results of this paper depend on the following lemmas. Lemma 1.1 (Cf. [6, Lemma 1.2.11]). Let {Xn , n ≥ 1} be a sequence of ψ -mixing random variables. Let X ∈ F1k , Y ∈ Fk∞ +n , E |X | < ∞, E |Y | < ∞. Then E |XY | < ∞ and

|EXY − EXEY | ≤ ψ(n)E |X |E |Y |.

(1.3)

Lemma 1.2 (Cf. [3, Theorem 2.1] and [7, Lemma 1.5]). Let b1 , b2 , . . . be a nondecreasing unbounded sequence of positive numbers and α1 , α2 , . . . be nonnegative numbers. Let r and C be fixed positive numbers. Assume that for each n ≥ 1,

r n X E max |Sl | ≤C αl , 

1≤l≤n

∞ X αl

< ∞,

brl

l =1

(1.4)

l =1

(1.5)

then Sn

lim

n→∞

= 0 a.s.,

bn

(1.6)

and with the growth rate Sn bn

 =O

βn



a.s.,

bn

(1.7)

where δ/r

βn = max bk vk , 1≤k≤n

∀ 0 < δ < 1,

vn =

∞ X αk

brk k=n

,

lim

n→∞

βn bn

= 0.

(1.8)

In addition,

r n X Sl αl E max < ∞, ≤ 4C 1≤l≤n b br 

l

(1.9)

l

l =1

r ∞ X Sl αl ≤ 4C < ∞. E sup r b b l≥1 

l

l =1

(1.10)

l

If further we assume that αn > 0 for infinitely many n, then

 E

r ∞ X Sl αl ≤ 4C < ∞. β βr

sup l≥1

l

l =1

(1.11)

l

Lemma 1.3 (Cf. [3, Corollary 2.1], and [8, Corollary 2.1.1]). Let b1 , b2 , . . . be a nondecreasing unbounded sequence of positive Pk numbers and α1 , α2 , . . . be nonnegative numbers. Denote Λk = i=1 αi for k ≥ 1. Let r and C be fixed positive numbers satisfying (1.4). If ∞ X l =1

Λl



1 brl

−

1 brl+1



<∞

(1.12)

and

Λn brn

is bounded,

(1.13)

then (1.6)–(1.11) hold. Lemma 1.4 (Cf. [9, Lemma 2.2]). Let {Xn , n ≥ 1} be a sequence of ϕ -mixing random variables. Put Ta (n) = that there exists an array {Ca,n } of positive numbers such that ETa2 (n) ≤ Ca,n

for every a ≥ 0 and n ≥ 1.

Pa+n

i=a+1

Xi . Suppose (1.14)

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X. Wang et al. / Applied Mathematics Letters 23 (2010) 1156–1161

Then for every q ≥ 2, there exists a constant C depending only on q and ϕ(·) such that



max |Ta (j)|q

E



1≤j≤n

  ≤ C Caq,/n2 + E max

a+1≤i≤a+n

|Xi |q

 (1.15)

for every a ≥ 0 and n ≥ 1. 2. Maximal inequality for ψ -mixing sequences Theorem 2.1. Let {Xn , n ≥ 1} be a sequence of ψ -mixing random variables satisfying n=1 ψ(n) < ∞. q ≥ 2. Assume that EXn = 0 and E |Xn |q < ∞ for each n ≥ 1. Then there exists a constant C depending only on q and ψ(·) such that

P∞

 q ! a+j a +n X X E max Xi ≤C E |Xi |q + 1≤j≤n i=a+1 i=a+1

a+n X

EXi2

!q/2  

(2.1)

i=a+1

for every a ≥ 0 and n ≥ 1. In particular, we have

 q ! !q / 2  j n n X X X  Xi ≤C E max E |Xi |q + EXi2 1≤j≤n i=1 i=1 i=1

(2.2)

for every n ≥ 1. Proof. By Lemma 1.1, we can see that a +n X

E

!2 Xi

=

i=a+1

a +n X

a +n X

a +n X

a +n X

ψ(j − i)(EXi2 )1/2 (EXj2 )1/2

X

EXi2 + 2

a+1≤i
EXi2 +

i=a+1

≤

ψ(j − i)E |Xi |E |Xj |

a+1≤i
i=a+1

≤

X

EXi2 + 2

i=a+1

≤

E (Xi Xj )

a+1≤i
i=a+1

≤

X

EXi2 + 2

1+2

n −1 a + n−k X X

ψ(k)(EXi2 + EXk2+i )

k=1 i=a+1

∞ X

! ψ(k)

k=1

a +n X

.

EXi2 = C1

i=a+1

a+n X

EXi2 .

i=a+1

It is well known that ψ -mixing is also ϕ -mixing. Therefore, by Lemma 1.4, we can get the desired result (2.1) immediately. The proof is complete.  3. Hájek–Rényi-type inequality for ψ -mixing sequences Theorem 3.1. Let {Xn , n ≥ 1} be a sequence of ψ -mixing random variables satisfying nondecreasing sequence of positive numbers. Then for any ε > 0 and any integer n ≥ 1,

P∞

) j n 1 X 8C X Var(Xj ) P max (Xi − EXi ) ≥ ε ≤ 2 , 1≤j≤n bj ε j=1 b2j j =1

n =1

ψ(n) < ∞ and {bn , n ≥ 1} be a

(

(3.1)

where C is defined in Theorem 2.1. Proof. (2.2) in Theorem 2.1 implies that



2  j n X X E  max (Xi − EXi )  ≤ 2C Var(Xj ). 1≤j≤n i=1 j=1

(3.2)

Therefore, (3.1) follows from Markov’s inequality, (3.2) above and Theorem 1.1 of Fazekas and Klesov [3] immediately.



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Theorem 3.2. Let {Xn , n ≥ 1} be a sequence of ψ -mixing random variables satisfying n=1 ψ(n) < ∞ and {bn , n ≥ 1} be a nondecreasing sequence of positive numbers. Then for any ε > 0 and any positive integers m < n,

P∞

! j 1 X 32C P max (Xi − EXi ) ≥ ε ≤ 2 m≤j≤n bj ε j =1

m X Var(Xj ) j =1

b2m

+

n X Var(Xj )

b2j

j=m+1

! ,

(3.3)

where C is defined in Theorem 2.1. Proof. (2.1) in Theorem 2.1 implies that



2  j m X X E  max (Xi − EXi )  ≤ 2C Var(Xj ) k≤j≤m i =k j =k

(3.4)

for any positive integers k and m with 1 ≤ k < m ≤ n. Therefore, (3.3) follows from Markov’s inequality, (3.4) above and Theorem 2.3 of Fazekas [4] immediately.  Theorem 3.3. Let {Xn , n ≥ 1} be a sequence of ψ -mixing random variables satisfying n=1 ψ(n) < ∞ and {bn , n ≥ 1} be a Pn nondecreasing sequence of positive numbers. Denote Tn = j=1 (Xj − EXj ) for n ≥ 1. Assume that

P∞

∞ X Var(Xj )

< ∞,

b2j

j=1

(3.5)

then for any r ∈ (0, 2),

 E

r  ∞ X Tn Var(Xj ) ≤ 1 + 8Cr < ∞, 2 − r j =1 b2j n≥1 bn

sup

(3.6)

where C is defined in Theorem 2.1. Furthermore, if limn→∞ bn = +∞, then lim

n→∞

n 1 X

b n j =1

(Xj − EXj ) = 0 a.s.

(3.7)

Proof. By the continuity of probability and Theorem 3.1, we get

 E

r  Z Tn = n≥1 b

sup

n

r  Tn > t dt n≥1 bn 0   Z ∞ Tn ≤ 1+ lim P max > t 1/r dt N →∞ 1≤n≤N b ∞



P

sup

n

1

∞ 8Cr X Var(Xj ) ≤ 1+ < ∞, 2 − r j=1 b2j

r ∈ (0, 2),

which implies (3.6). By (3.4) above and Theorem 2.3 of Fazekas [4], we can see that

2 ! Tk E sup = lim E n→∞ k≥m b k

! 2 ! m ∞ X X Tk Var(Xj ) Var(Xj ) max ≤ 32C + . m≤k≤n b b2 b2 k

j=1

m

j=m+1

(3.8)

j

Combining (3.5) with (3.8), we have

2 ! Tk lim E sup = 0, m→∞ k≥m b

(3.9)

k

which implies (3.7). The proof is complete.



4. Strong law of large numbers and growth rate for ψ -mixing sequences Theorem 4.1. Let {Xn , n ≥ 1} be a sequence of mean zero ψ -mixing random variables satisfying {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers. Assume that

P∞

n=1

ψ(n) < ∞ and

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X. Wang et al. / Applied Mathematics Letters 23 (2010) 1156–1161

∞ X EX 2

< ∞,

n

b2n

n =1

(4.1)

then (1.6)–(1.11) hold, where r = 2, αk = 2EXk2 , k ≥ 1 and C is defined in Theorem 2.1. Proof. By (2.2) in Theorem 2.1, we have



2



max |Sk |

E

≤ 2C

1≤k≤n

n X

n X

EXi2 = C

i=1

αk .

(4.2)

k=1

It follows by (4.1) that ∞ X αn n =1

=2

b2n

∞ X EX 2 n

n=1

b2n

< ∞.

(4.3)

Thus, (1.6)–(1.11) follow from (4.2) and (4.3) and Lemma 1.2 immediately.



Theorem 4.2. Let {Xn , n ≥ 1} be a sequence of ψ -mixing random variables satisfying n ≥ 1. 1 ≤ p < 2. Denote Qn = max1≤k≤n EXk2 for n ≥ 1 and Q0 = 0. Assume that ∞ X Qn n =1

P∞

n =1

ψ(n) < ∞ and EXn2 < ∞ for

< ∞,

n2/p

(4.4)

then lim

n 1 X

n→∞

n1/p i=1

(Xi − EXi ) = 0 a.s.,

(4.5)

and with the growth rate n 1 X

n1/p i=1

(Xi − EXi ) = O

βn





n1/p

a.s.,

(4.6)

where δ/2

βn = max k1/p vk ,

∀0 < δ < 1,

1≤k≤n

vn =

∞ X αk k=n

αk = 2kQk − 2(k − 1)Qk−1 ,

k ≥ 1,

lim

n→∞

k2/p

βn n1/p

,

= 0.

(4.7)

In addition, (1.9)–(1.11) hold, where r = 2, bn = n1/p and C is defined in Theorem 2.1. Furthermore, for any r ∈ (0, 2),

r 



 E

Sl sup 1/p l ≥1 l

≤1+

∞ 4Cr X αl

2 − r l=1 l2/p

< ∞.

(4.8)

l=1 αl , n ≥ 1. By (2.2) in Theorem 2.1, we can see that  2 k n n X X X  E max Xi  ≤ 2C EXi2 ≤ 2CnQn = C αk . 1≤k≤n i=1 i =1 k=1

Proof. Assume that EXn = 0 and Λn =

Pn



(4.9)

It is a simple fact that αk ≥ 0 for all k ≥ 1. It follows by (4.4) that ∞ X l =1

Λl

1 b2l

−

1 b2l+1

! =2

∞ X l =1

 lQl

1 l2/p

−

1

(l + 1)2/p

 ≤

∞ 4 X Ql

p l=1 l2/p

< ∞.

(4.10)

That is to say (1.12) holds. By Remark 2.1 in Fazekas and Klesov [3], (1.12) implies (1.13). By Lemma 1.3, we can obtain (4.5)–(4.7) and (1.9)–(1.11) immediately. Similar to the proof of (3.6) in Theorem 3.3, it is easy to get (4.8). The proof is complete. 

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Acknowledgements The authors are most grateful to the Editor-in-Chief Ervin Y. Rodin and two anonymous referees for careful reading of the manuscript and valuable suggestions which helped to significantly improve an earlier version of this paper. References [1] J.R. Blum, D.L. Hanson, L. Koopmans, On the strong law of large numbers for a class of stochastic process, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 2 (1963) 1–11. [2] S.C. Yang, Almost sure convergence of weighted sums of mixing sequences, J. Systems Sci. Math. Sci. 15 (3) (1995) 254–265 (in Chinese). [3] I. Fazekas, O. Klesov, A general approach to the strong law of large numbers, Theory Probab. Appl. 45 (2001) 436–449. [4] I. Fazekas, On a general approach to the strong laws of large numbers. 2006. http://www.inf.unideb.hu/valseg/dolgozok/fazekasi/preprintek.html. [5] T. Tómács, Zs. Líbor, A Hájek–Rényi type inequality and its applications, Ann. Math. Inform. 33 (2006) 141–149. [6] C.R. Lu, Z.Y. Lin, Limit Theory for Mixing Dependent Sequences, Science Press, Beijing, China, 1997 (in Chinese). [7] S.H. Hu, G.J. Chen, X.J. Wang, On extending the Brunk–Prokhorov strong law of large numbers for martingale differences, Statist. Probab. Lett. 78 (2008) 3187–3194. [8] S.H. Hu, Some new results for the strong law of large numbers, Acta Math. Sinica 46 (6) (2003) 1123–1134 (in Chinese). [9] Q.M. Shao, Almost sure invariance principles for mixing sequences of random variables, Stochastic Process. Appl. 48 (1993) 319–334.