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## Academic research paper on topic "Strong convergence properties for ψ-mixing random variables"

﻿0 Journal of Inequalities and Applications

a SpringerOpen Journal

RESEARCH

Open Access

Strong convergence properties for ty-mixing random variables

Huai Xu1 and Ling Tang2

Correspondence: tang_ling_xu@126.com 2Department of Mathematics and Physics, AnhuiJianzhu University, Hefei, 230022, P.R. China Fulllist of author information is available at the end of the article

Abstract

In this paper, by using the Rosenthal-type maximal inequality for ft-mixing random variables, we obtain the Khintchine-Kolmogorov-type convergence theorem, which can be applied to establish the three series theorem and the Chung-type strong law of large numbers for ft-mixing random variables. In addition, the strong stability for weighted sums of ft-mixing random variables is studied, which generalizes the corresponding one of independent random variables. MSC: 60F15

Keywords: strong stability; Khintchine-Kolmogorov-type convergence theorem; ft-mixing random variables

ft Spri

ringer

1 Introduction

Let (fi, F, P) be a fixed probability space. The random variables we deal with are all defined on (fi, F,P). Throughout the paper, let I(A) be the indicator function of the set A. For random variable X, denote X(c) = XI(|X| < c) for some c > 0. Denote log+ x = lnmax(e,x). C and c denote positive constants, which may be different in various places.

Let (Xn, n > 1} be a sequence of random variables defined on a fixed probability space (fi, F, P), and let Sn = ^"i=1 Xi for each n > 1. Let n and m be positive integers. Write Fm = a (Xi, n < i < m). Given a-algebras B, R in F, let

,<n-p\ |P(AB)-P(A)P(B)|

ft(B, R) = sup -PA)P(B)-• (1-1)

AeE,BeRP(A)P(B)>0 P(A)P(B)

Define the mixing coefficients by

f (n) = sup f F, F£n), n > 0.

Definition 1.1 A sequence (Xn, n > 1} of random variables is said to be a sequence of ft-mixing random variables if ft(n) | 0 as n ^ro.

The concept of ft-mixing random variables was introduced by Blum etal. [1] and some applications have been found. See, for example, Blum et al. [1] for strong law of large numbers, Yang [2] for almost sure convergence of weighted sums, Wu [3] for strong consistency of M estimator in linear model, Wang et al. [4] for maximal inequality and Hajek-Renyi-type inequality, strong growth rate and the integrability of the supremum, Zhu et al. [5] for

© 2013 Xu and Tang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

strong convergence properties, Pan etal. [6] for strong convergence of weighted sums, and so on. 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 establish the Khintchine-Kolmogorov-type convergence theorem, which can be applied to obtain the three series theorem and the Chung-type strong law of large numbers for ty-mixing random variables. In addition, we will study the strong stability for weighted sums of ty-mixing random variables, which generalizes the corresponding one of independent random variables.

For independent and identically distributed random variable sequences, Jamison et al. [7] proved the following theorem.

TheoremA Let {X, Xn, n > 1} be an independent and identically distributed sequence with the same distribution function F (x), and let {wn, n > 1} be a sequence of positive numbers. Write Wn = £ti wn and N(x) = Card{n : WJun < x}, x > 0. If

(i) Wn and Mn W— ^ 0 as n

(ii) E|X| < co and EN(|X|) < c,

(iii) XCCCx2(/y>|x| N(y)/y3 dy) dF(x) < c, then

1 ^ MiXi ^ c

where c is a constant.

The result of Theorem A for independent and identically distributed sequences has been generalized to some dependent sequences, such as negatively associated sequences, negatively superadditive dependent sequences, p-mixing sequences, ip-mixing sequences, and so forth. We will further study the strong stability for weighted sums of ty-mixing random variables, which generalizes corresponding one of independent sequences. The main results of the paper depend on the following important lemma - Rosenthal-type maximal inequality for ty-mixing random variables.

Lemma 1.1 (cf. Wang etal. [4]) Let {Xn, n > 1} be a sequence of ty-mixingrandom variables satisfying £CC=i ty (n)< c, q > 2. Assume that EXn = 0 and EX^ < c for each n > 1. Then there exists a constant C depending only on q and ty (■) such that

E( max

Y^E^ + £ EXf

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

\ 1<;<n

^EXY + £ EX2

for every n > 1.

The following concept of stochastic domination will be used frequently throughout the paper.

Definition 1.2 A sequence {Xn, n > 1} of random variables is said to be stochastically dominated by a random variable X if there exists a constant C such that

P(|Xni > x) < CP(|X| > x) (1.5)

for all x > 0 and n > 1.

By the definition of stochastic domination and integration by parts, we can get the following basic property for stochastic domination. For the proof, one can refer to Wang et al. [8], Tang [9] or Shen and Wu [10].

Lemma 1.2 Let {Xn, n > 1} be a sequence of random variables, which is stochastically dominated by a random variable X. For any a >0 and b >0, the following statement holds

EIXnial(iXn| < b) < C{EiXial(|X| < b) + baP(|X| > b)}, where C is a positive constant.

2 Khintchine-Kolmogorov-type convergence theorem

In this section, we will prove the Khintchine-Kolmogorov-type convergence theorem for ft-mixing random variables. By using the Khintchine-Kolmogorov-type convergence theorem, we can get the three series theorem and the Chung-type strong law of large numbers for ft-mixing random variables.

Theorem 2.1 (Khintchine-Kolmogorov-type convergence theorem) Let {Xn, n > 1} be a sequence of ft -mixing random variables satisfying X^i ft (n) < ro. Assume that

^Var (Xn)<ro, (.1)

then J^ro 1(Xn - EXn) converges a.s.

Proof Without loss of generality, we assume that EXn = 0 forall n > 1. For any e > 0, it can be checked that

P( sup |Sk - SmI > e) < P( sup |Sk - Sn| > e) + P( sup |Sm - Sn | >

k,m>n ' \k>n 2' \m>n

< 2 lim PI max |Sk - Sn | >

N^ro \n<k<N 2

< 2 lim — V Var(Xi)

N(§)2 i=n+1

— J^ Var (Xi) ^ 0, n ^ ro,

where the last inequality follows from Lemma 1.1. Thus, the sequence {Sn, n > 1} is a.s. Cauchy, and, therefore, we can obtain the desired result immediately. This completes the proof of the theorem. □

With the Khintchine-Kolmogorov-type convergence theorem in hand, we can get the three series theorem and the Chun-type strong law oflarge numbers for ty-mixing random variables.

Theorem 2.2 (Three series theorem) Let {Xn, n > 1} be a sequence of ty-mixing random variables satisfying ty(n) < ro. For some c > 0, if

J2P(\Xn\> c) < ro, (2.2)

J^EX^ converges, (2.3)

^Var(X<c)) < ro, (2.4)

then £ro=1 Xn converges almost surely.

Proof According to (2.4) and Theorem 2.1, we have

- EX(n]) converges a.s. (2.5)

It follows by (2.3) and (2.5) that

^Xnc) converges a.s. (2.6)

Obviously, (2.2) implies that

J^P(Xn = Xnr) = J2P(\Xn\> c) < ro. (2.7)

n=1 n=1

It follows by (2.7) and Borel-Cantelli lemma that

P(Xn = X^i.o.) = 0. (2.8)

Finally, combining (2.6) with (2.8), we can get that £J^aXn converges a.s. The proof is completed. □

Theorem 2.3 (Chung-type strong law of large numbers) Let {Xn, n > 1} be a sequence of mean zero ty-mixing random variables satisfying ty(n) < ro, and let {an, n > 1} be a sequence of positive numbers satisfying 0 < an fro. If there exists some p e [1,2] such that

ro E\Xn\p

< ro, ()

lim — V Xi = 0 a.s. (2.10)

n-ro an i=1

Proof It follows by (2.9) that

^ Var(X^< v E(x«n))2

n=1 n n=1

= ^ EXll(\Xn\<an) ^ a2

n=1 an

ro E\Xn\p

IT <ro.

Therefore, we have by Theorem 2.1 that

ro v-(an) rv(an)

EXn - EXn

- converges a.s. (2.11)

Since p e [1,2], it follows by EXn = 0 that

ro (an) ro S^ \EXn \ Ty^

\EXnn)\ ro\EXnI (\Xn\<an)\

n=1 an n=1 an

= ^ \EXnI(\Xn\ > an)\

n=1 an

ro E| Xn | p

^T <ro,

n=1 an

which implies that

ro EX^an)

y^ —n— converges. (2.12)

n=1 an

Together with (2.11) and (2.12), we can see that

ro X(an)

y^ —— converges a.s. (2.13)

n=1 an

By Markov's inequality and (2.9), we have

ro ( ) ro ( ) ro p

J^P(Xn = X(an)) = £P(\Xn\ > an) < £ ^ < ro. (2.14)

Hence, the desired result (2.10) follows from (2.13), (2.14), Borel-Cantelli lemma and Kronecker's lemma immediately. □

3 Strong stability for weighted sums of rf -mixing random variables

In the previous section, we were able to get the Khintchine-Kolmogorov-type convergence theorem for ft-mixing random variables. In this section, we will study the strong stability for weighted sums of ft-mixing random variables by using the Khintchine-Kolmogorov-type convergence theorem. The concept of strong stability is as follows.

Definition 3.1 A sequence {Yn, n > 1} is said to be strongly stable if there exist two constant sequences {bn, n > 1} and {dn, n > 1} with 0 < bn tTO such that

b~^Yn - dn ^ 0 a.s.

For the definition of strong stability, one can refer to Chow and Teicher [11]. Many authors have extended the strong law of large numbers for sequences of random variables to the case of triangular array of rowwise random variables and arrays of rowwise random variables. See, for example, Hu and Taylor [12], Bai and Cheng [13], Gan and Chen [14], Kuczmaszewska [15], Wu [16-18], Sung [19], Wang et al. [20-24], Zhou [25], Shen [26], Shen etal. [27], and so on. Our main results are as follows.

Theorem 3.1 Let {an, n > 1} and {bn, n > 1} be two sequences of positive numbers with cn = bn /an and bn f to. Let {Xn, n > 1} be a sequence of ft -mixing random variables, which is stochastically dominated by a random variable X. Assume that ETO ft (n) < to. Denote N(x) = Card{n : cn < x}, x > 0,1 < p < 2. If the following conditions are satisfied

(i) EN(|X|) < to,

(ii) /0TO tp-1P(|X| > t)(/tTON(y)/yp+1 dy) dt < to, then there exist dn e R, n = 1,2,..., such that

b-1 ^ aiXi - dn ^ 0 a.s. (3.1)

Proof Let Sn = En=i aiXi, Tn = En=i aiX(c'). By Definition 1.2 and (i), we can see that

TO TO TO

J^P(Xi = X(ci ) = £ P( |Xi| > c) < cY^P{ |X| > *) < CEN( |X|) < TO. (3.2)

i=1 i=1 i=1

By Borel-Cantelli lemma, for any sequence {dn, n > 1}C R, the sequences {bn1Tn - dn} and {b^1Sn - dn} converge on the same set and to the same limit. We will show that b^En=i ai{x(ci) - Ex(ci)) ^ 0 a.s., which gives the theorem with dn = b^Eti aiExfi). Note that {ai(x!f'i - EX(c,)), i > 1} is a sequence of mean zero ft-mixing random variables. It follows from Cr inequality, Jensen's inequality and Lemma 1.2 that

>E|an(Xncn)- EX""^

< C^c-pE(|Xn|pI(|Xn|<cn))

< ^c-p[cpP(\X\ > cn) + E\X\pI(\X\ < c«)}

ro ( ) ro cn ( )

< C£p(\X\ > cn) + c^« / tp-1P( \X\ > t)dt (3.3)

n=1 n=1

ro cn ( ) ro ( )

V c«p / t?-1P(\X\ > t)dt < tp-1P(\X\ > t)J2 « dt

n=1 J° J0 n:c« >t

< C^ tp-1P(\X\ > t)Q" N(y)/yp+1 d^jdt. (3.4) The last inequality above follows from the fact that

E c«p = ^ E c«p

n:c«>t n:t<c«<u

lim y-pdN (y)

u—rot/ t

H—n ^u-pN(u) - t-pN(t) + p £ y-(p+1)N(y) dy^

' — 0 as u —^ ro.

u-pN(u) < p / y-(p+1)N(y) dy — 0

Obviously,

£>(\X\ > cn) < EN(\X\) < ro. (.5)

Thus, by (3.3)-(3.5) and condition (ii), we can see that

^ E\a«(X«cn) -EX(ncn))\p

Y-Jp-< ro. ((6)

n=1 bn

Therefore,

b-1£ ai(X(ci)- EX(ci)) — 0 a.s.,

following from (3.6), Theorem 2.3 and Kronecker's lemma immediately. The desired result is obtained. □

Corollary 3.1 Let the conditions of Theorem 3.1 be satisfied, and let EXn = 0 for n > 1. Assume that /j_ro EN(\X\/s) ds < ro. Then b«1^,«=i aiXi — 0 a.s.

Proof By Theorem 3.1, we only need to prove that

b-1J2 aiEXfi) — 0 a.s. (3.7)

In fact,

TO n IFy(ci)\ TO TO

Y^^ElA = £ c^EX^ < ci)| < £ ^Emid^ > ci) bi

i=1 i=1 i=1

TO / /»TO \

- £ c-^|Xi | > c) + j P( IXi | > t) dtj

EN(IXI/s)ds < to,

which implies (3.7) by Kronecker's lemma. We complete the proof of the corollary. □

Theorem 3.2 Let {an, n > 1} and {bn, n > 1} be two sequences of positive numbers with cn = bn/an and bn t to. Let {Xn, n > 1} be a sequence of mean zero ft -mixing random variables, which is stochastically dominated by a random variable X. Assume thatYf^ ft(n) < to. Denote N(x) = Card{n: cn — x}, x > 0,1 — p — 2. If the following conditions are satisfied

(i) EN(|X|) < to,

(ii) _/1TOEN(|X|/s) ds < to,

(iii) max1—j— n dp E,TOn c-p = O(n), then

b-1J2 aiXi ^ 0 a.s. (3.8)

Proof By condition (i) and (3.2), we only need to prove that bn1J2n=1 aiX(i) ^ 0 a.s. For this purpose, it suffices to show that

b-^Y,ai{X(ci)-EX^) ^ 0 a.s. (3.9)

b-1J2 aiEXfi) ^ 0 as n ^to. (.10)

Equation (3.10) follows from the proof of Corollary 3.1 immediately.

To prove (3.9), we set e0 = 0 and en = max1</—n cj for n > 1. It follows from Cr inequality, Jensen's inequality and Lemma 1.2 that

TO Fl^r iv(cn) pv(cn)\I p TO

EE|an(Xn ' n )| — C^fE^I (|Xn| — cn))

n=1 bn n=1

— |X| > n + Cj2c7EX|pl(|X| — n.

n=1 n=1

Obviously,

£]P(|X| > cn) — EN(|X|) < TO (3.11)

J2c7E\X\pI(|X| < cn) <J2c-npE\X\pI(|X| < sn)

n=l n=l

< E efP^-i < |X | < jJ2cnP < CJ2P( X | >

j=l n=i i=l

< C^l + YHX| > Cn^^ < c(l +EN(|X|)) < TO.

Therefore,

^ E|fln(XnCn)- EXi^P

Y-TP-< TO, ((2)

following from the statements above. By Theorem 2.3 and Kronecker's lemma, we can obtain (3.9) immediately. The proof is completed. □

Theorem 3.3 Let {an, n > l} and {bn, n > l} be two sequences of positive numbers with cn = bnlan and bn \ to. Let {Xn, n > l} be a sequence of ty-mixing random variables, which is stochastically dominated by a random variable X. Assume that XTO^ ty (n) < to. Define N(x) = Card{n: cn < x}, R(x) = /TO N(y)y-3 dy, x > 0. If the following conditions are satisfied

(i) N (x) < to for any x >0,

(ii) R(l) = fTO N(y)y-3 dy < to,

(iii) EX2R(|X|)<to,

then there exist dn e R, n = l, 2,..., such that

b-1^^ aiXi - dn ^ 0 a.s. (3.l3)

Proof Since N(x) is nondecreasing, then for any x >0 fTO l

R(x) > N(x) y-3 dy = -x-2N(x), (3.l4)

which implies that EN(|X|) < 2EX2R(|X|) < to. Therefore,

J2P(Xi = X(ci)) = £P(|Xi| > ci)

i=l i=l

< c£p(|X| > ci) < CEN(m) < TO. (3.l5)

By Borel-Cantelli lemma for any sequence {dn, n > l}C R, {b-l5n - dn} and {b-1Tn - dn} converge on the same set and to the same limit. We will show that b-1J2n=l ai(X(,) -EX(ci)) ^ 0 a.s., which gives the theorem with dn = b-lXn=l aiEX(c'). It follows from

Lemma 1.2 that

Vow« v^nh to to

^WtaXj < ^^E^)2 = £c-2£X;2/(X| <

n=1 n n=1 n=1

< CEN(|X|) + C^^ c-2EX2I(|X| < cn) (3.16)

J2cn2EX2l(|X| — cn) = J2 cn2EX2I(|X| — cn) +J2 cn2EX2l{|X| — cn)

n=1 n:cn— 1 n:c«>1

= I1+12. (7)

Since N(1) = Card{n : cn — 1} — 2R(1) < to from (3.14) and (ii), it follows that I1 < to. For I2, we have

I2= £ cn,2EX2I( |X| < cn) = £ £ cn,2EX2I( |X| < ,

n:cn>1 k=2 k-1<cn<k

< J2(N(k) - N(k - 1)) (k -1)-2EX2I(|X| < 1)

+ ^(N(k) - N(k - 1))(k - 1)-2EX2I(1 < |X| < k)

= I21 + I22>

to to to 7+1

I21 < ^(N (k) - N (k -1)) £ 7-3 = Cj2f3J2 (N(k) - N (k -1))

k=2 7=k-1 7=1 k=2

TO /»TO

— C J2 (j + 1)-3N (j + 1) — C y-3N (y) dy < to.

Since N(x) is nondecreasing and R(x) is nonincreasing, we have

I22 — J2 EX21 (m -1 < |X|— m)Yl N (k)((k -1)-2 - k-2)

m=2 k=m

TO TO /. k+1

— Cj2EX2I(m -1 < |X|— m) ^ / N(x)x-3 dx

m=2 k=m^k

— C^EX2R(|X|)I(m -1< |X| — m) — CEX2R(|X|) < to.

Therefore,

^ Var(anXncn))

L—b5— <TO (3.18)

following from the above statements. By Theorem 2.1 and Kronecker's lemma, we have

b-1J2 «(X^ - EX(C']) — 0 a.s. (3.19)

Taking dn = bn1J2n=1 a;EX(c,), we have bn1J2n=1 aiX(,) - dn — 0 a.s. The proof is completed. □

Corollary 3.2 Let the conditions of Theorem 3.3 be satisfied. If EXn = 0, n > 1 and

EW(|X|/s) ds < TO, then b-1 £n=1 aiXi — 0 a.s.

In the following, we denote a(x) : R+ — R+ as a positive and nonincreasing function with an = a(n), bn = Xn=1 ai, cn = bn/an, n > 1, where

0 < bn f to, (.20)

0 < liminfn-1cna(logcn) < limsupn-1cna(logcn) < to, (3.21)

n ^TO n—>TO

xa(log+ x) is nonincreasing for x >0. (3.22)

Theorem 3.4 Let {Xn, n > 1} be a sequence of identically distributed ty-mixing random variables with^2TO=i_ ty (n) < to. IfE|X1|a(log+ |Xi|)< to, then there exist dn e R, n = 1,2,..., such that b-1 Xn=1 aiXi - dn — 0 a.s.

Proof Since a(x) is positive and nonincreasing for x >0 and 0 < bn fTO, it follows that cn f to. By (3.21), we can choose constants m e N, C1 > 0, C2 > 0 such that for n > m,

C1n < cna(log cn) < C2n. (.3)

Therefore, for n > m,we have -1 < a(l°gcm), which implies that

cn C1 n

-2 ^ ^ a2(log cm) _ a2(log cm) iQ

j=m j=m 1 1

By (3.22)-(3.24), it follows that

1 E(a>X<c))2 ^ f X2 dp + Y I x2 dp

Lj^ cflj xUP + Yf

j=m bj i=m \ {|X1| <cm-1} i=mJ {ci-1

j j=m y{|X1|<cm-1} ¡=mJ {ci-l<|Xl|<Ci}

TO j r

< C + £ cfY X2 dP

j=m i=m ^ {ci-1<|X1|<ci}

C + C J2 i-1a2(log ci) X2 dP

i=m J {ci-1<|Xl|<Ci}

TO „

C + C^ a (log ci) |X1| dP

i=m ^{c,'-1<|X1|<c;}

< C + C^ |X1|^log+ |X^) dP < TO.

i=m^ {c;-1<|X1|<c;}

Therefore,

~ Var(a;-X; ) ~ E(aX j))2 , , E-ir— — E -j-< TO, (5)

j=1 bj2 j=1 bj2

which implies that

b-1^ a^xf - EX(c)) ^ 0 a.s. (3.26)

from Theorem 2.1 and Kronecker's lemma. By (3.22) and (3.23) again, we have

£p( |Xj| > j — J2 P( |Xj|^log+ |Xj|) > cj a (log cj))

j= m j= m

— EP{|X1|^log+ |X1^ > Cj < to,

to to m-1 to

J2P(x =Xj = E P( x > j = E P( x > j + E p{ X > j < TO.

j=1 j=1 j=1 j=m

By Borel-Cantelli lemma, we have P(Xj = Xj' ,i.o.) = 0. Together with (3.26), we can see that

b-1J2 a(Xi - Exfi]) ^ 0 a.s. (3.27)

Taking dn = b-1 En=1 aiEX\c' for n > 1, we get the desired result. □

Theorem 3.5 Let {Xn, n > 1} be a sequence of ft -mixing random variables with ETO=1 ft (n) < to. If for some 1 — p — 2,

J2n~pE\Xna(log+ X»|)|p < to,

then there exist dn e R, n = 1,2,..., such that b-1 Et=1 aiXi - dn ^ 0 a.s. Proof Similar to the proof of Theorem 3.4, it is easily seen that

J2HXj = X(i) — m - 1 + £P(|Xj|a(log+ j > c,a(logc,))

j=1 j= m

— m -1 + EP(|Xj|a(log+ |Xj0 > Cj < to.

By Borel-Cantelli lemma for any sequence {dn, n > 1}C R, the sequences {b- En=1 aiXi -dn} and {b-1 En=1 aXf - dn} converge on the same set and to the same limit. We will show

that b-1 Xn=1 ai(Xifi) -EXifi)) — 0 a.s., which gives the theorem with dn = b-1 ^n=1 aiEx(i). Note that {ai(X(c') - EX(c'))/bi, i > 1} is a sequence of mean zero ty-mixing random variables. By Cr inequality and Jensen's inequality, we can see that

,E\aj(xjCi)- ExjCi))\P

E j j j-< cm -1) + C £c-pE\Xj\p^\XI < Cj)

j=1 j j=m

< C(m -1) + C^r(«(logCj))pE\Xj\Pl(\Xj\ < Cj)

< C(m -1) + Cj2j~PE\Xja(log+ \X;\) \p < œ.

It follows by Theorem 2.3 that b-1 X"=1 ai (X(Ci) -EX(Ci)) ^ 0 a.s. The proof is completed.

Corollary 3.3 Let the Conditions of Theorem 3.5 be satisfied. Furthermore, suppose that EXn = 0 and Xœ=1 /œ P(\Xn \ > sCn) ds < œ, then b-1 Xn=1 aiXi ^ 0 a.s.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors read and approved the finalmanuscript Author details

1 Schoolof MathematicalScience, Anhui University, Hefei, 230601, P.R. China. 2 Department of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230022, P.R. China.

Acknowledgements

The authors are most gratefulto the editor Andrei Volodin and an anonymous referee for carefulreading of the manuscript and valuable suggestions, which helped in improving an earlier version of this paper. This work was supported by the NaturalScience Project of Department of Education of Anhui Province (KJ2011z056) and the National NaturalScience Foundation of China (11201001).

Received: 23 April 2013 Accepted: 24 July 2013 Published: 2 August 2013 References

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doi:10.1186/1029-242X-2013-360

Cite this article as: Xu and Tang: Strong eonvergenee properties for ft-mixing random variables. Journal of Inequalities and Applications 2013 2013:360.

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