**R E S E A R C H**

**Open Access**

## Strong convergence properties for

*ψ*

## -mixing

## random variables

### Huai Xu

1_{and Ling Tang}

2*
*_{Correspondence:}

2_{Department of Mathematics and}

Physics, Anhui Jianzhu University, Hefei, 230022, P.R. China Full list of author information is available at the end of the article

**Abstract**

In this paper, by using the Rosenthal-type maximal inequality for

*ψ*

-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*ψ*

-mixing random variables. In addition, the strong stability for
weighted sums of*ψ*

-mixing random variables is studied, which generalizes the
corresponding one of independent random variables.
**MSC:** 60F15

**Keywords:** strong stability; Khintchine-Kolmogorov-type convergence theorem;

*ψ*

-mixing random variables
**1 Introduction**

Let (,*F*,*P*) be a ﬁxed probability space. The random variables we deal with are all deﬁned
on (,*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}_{> . Denote}_{log}+_{x}_{=}. _{ln max}_{(}_{e}_{,}_{x}_{).}

*C*and*c*denote positive constants, which may be diﬀerent in various places.

Let{*Xn*,*n*≥}be a sequence of random variables deﬁned on a ﬁxed probability space
(,*F*,*P*), and let*Sn*=*n _{i}*

_{=}

*Xi*for each

*n*≥. Let

*n*and

*m*be positive integers. Write

*Fm*

*n* =

*σ*(*Xi*,*n*≤*i*≤*m*). Given*σ*-algebras*B*,*R*in*F*, let

*ψ*(*B*,*R*) = sup

*A∈B*,*B∈R*,*P*(*A*)*P*(*B*)>

|*P*(*AB*) –*P*(*A*)*P*(*B*)|

*P*(*A*)*P*(*B*) . (.)

Deﬁne the mixing coeﬃcients by

*ψ*(*n*) =sup

*k≥*

*ψF*_{}*k*,*F _{k}*∞

_{+}

*,*

_{n}*n*≥.

**Deﬁnition .** A sequence{*Xn*,*n*≥} of random variables is said to be a sequence of

*ψ*-mixing random variables if*ψ*(*n*)↓ as*n*→ ∞.

The concept of*ψ*-mixing random variables was introduced by Blum*et al.*[] and some
applications have been found. See, for example, Blum*et al.*[] for strong law of large
num-bers, Yang [] for almost sure convergence of weighted sums, Wu [] for strong consistency
of*M*estimator in linear model, Wang*et al.*[] for maximal inequality and
Hájek-Rényi-type inequality, strong growth rate and the integrability of the supremum, Zhu*et al.*[] for

strong convergence properties, Pan*et al.*[] 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
pa-per 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
num-bers for*ψ*-mixing random variables. In addition, we will study the strong stability for
weighted sums of*ψ*-mixing random variables, which generalizes the corresponding one
of independent random variables.

For independent and identically distributed random variable sequences, Jamison*et al.*

[] proved the following theorem.

**Theorem A** *Let*{*X*,*Xn*,*n*≥}*be an independent and identically distributed sequence with*
*the same distribution function F*(*x*),*and let*{*wn*,*n*≥}*be a sequence of positive numbers*.

*Write Wn*=*n _{i}*

_{=}

*ωnand N*(

*x*) =Card{

*n*:

*Wn*/

*ωn*≤

*x*},

*x*> .

*If*(i)

*Wn*→ ∞

*andωnW*–

*n* →*asn*→ ∞,

(ii) *E*|*X*|<∞*andEN*(|*X*|) <∞,
(iii) _{–}∞_{∞}*x*(* _{y≥|x|}N*(

*y*)/

*y*

*dy*)

*dF*(

*x*) <∞,

*then*

*W _{n}*–

*n*

*i*=

*ωiXi*→*c* *a.s.*, (.)

*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,
nega-tively superadditive dependent sequences,*ρ*˜-mixing sequences,*ϕ*˜-mixing sequences, and
so forth. We will further study the strong stability for weighted sums of*ψ*-mixing random
variables, which generalizes corresponding one of independent sequences. The main
re-sults of the paper depend on the following important lemma - Rosenthal-type maximal
inequality for*ψ*-mixing random variables.

**Lemma .**(*cf.*Wang*et al.*[]) *Let*{*Xn*,*n*≥}*be a sequence ofψ-mixing random variables*
*satisfying*∞_{n}_{=}*ψ*(*n*) <∞,*q*≥.*Assume that EXn*= *and E*|*Xn*|*q*<∞*for each n*≥.*Then*
*there exists a constant C depending only on q andψ*(·)*such that*

*E*

max

≤j≤n

*a*+*j*

*i*=*a*+

*Xi*

*q*

≤*C*

_{a}_{+}_{n}

*i*=*a*+

*E*|*Xi*|*q*+
_{a}_{+}_{n}

*i*=*a*+

*EX _{i}*

*q*/

(.)

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

*E*

max

≤j≤n

*j*

*i*=

*Xi*

*q*

≤*C*

_{n}

*i*=

*E*|*Xi*|*q*+

_{n}

*i*=

*EX _{i}*

*q*/

(.)

*for every n*≥.

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

*P*|*Xn*|>*x*

≤*CP*|*X*|>*x* (.)

for all*x*≥ and*n*≥.

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

**Lemma .** *Let*{*Xn*,*n*≥}*be a sequence of random variables*,*which is stochastically *
*dom-inated by a random variable X*.*For anyα*> *and b*> ,*the following statement holds*

*E*|*Xn*|*αI*|*Xn*| ≤*b*≤*CE*|*X*|*αI*|*X*| ≤*b*+*bαP*|*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
*ψ*-mixing random variables. By using the Khintchine-Kolmogorov-type convergence
the-orem, we can get the three series theorem and the Chung-type strong law of large numbers
for*ψ*-mixing random variables.

**Theorem .**(Khintchine-Kolmogorov-type convergence theorem) *Let*{*Xn*,*n*≥}*be a*
*sequence ofψ-mixing random variables satisfying*∞_{n}_{=}*ψ*(*n*) <∞.*Assume that*

∞

*n*=

Var(*Xn*) <∞, (.)

*then*∞_{n}_{=}(*Xn*–*EXn*)*converges a*.*s*.

*Proof* Without loss of generality, we assume that*EXn*= for all*n*≥. For any*ε*> , it can

be checked that

*P*

sup

*k*,*m≥n*

|*Sk*–*Sm*|>*ε*

≤*P*

sup

*k≥n*|*Sk*–*Sn*|>

*ε*

+*P*

sup

*m≥n*|*Sm*–*Sn*|>

*ε*

≤ lim

*N→∞P*

max

*n≤k≤N*|*Sk*–*Sn*|>

*ε*

≤ lim

*N→∞*

(_{}*ε*)

*N*

*i*=*n*+

Var(*Xi*)

=
*ε*

∞

*i*=*n*+

Var(*Xi*)→, *n*→ ∞,

where the last inequality follows from Lemma .. Thus, the sequence{*Sn*,*n*≥}is a.s.
Cauchy, and, therefore, we can obtain the desired result immediately. This completes the

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

**Theorem .**(Three series theorem) *Let*{*Xn*,*n*≥}*be a sequence ofψ-mixing random*
*variables satisfying*∞_{n}_{=}*ψ*(*n*) <∞.*For some c*> ,*if*

∞

*n*=

*P*|*Xn*|>*c*<∞, (.)

∞

*n*=

*EX _{n}*(

*c*)

*converges*, (.)

∞

*n*=

Var*X _{n}*(

*c*)<∞, (.)

*then*∞_{n}_{=}*Xnconverges almost surely*.

*Proof* According to (.) and Theorem ., we have

∞

*n*=

*X _{n}*(

*c*)–

*EX*(

*) converges a.s. (.)*

_{n}cIt follows by (.) and (.) that

∞

*n*=

*X*(* _{n}c*) converges a.s. (.)

Obviously, (.) implies that

∞

*n*=

*PXn*=*X _{n}*(

*c*)=

∞

*n*=

*P*|*Xn*|>*c*<∞. (.)

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

*PXn*=*X _{n}*(

*c*), i.o.= . (.)

Finally, combining (.) with (.), we can get that∞_{n}_{=}*Xn*converges a.s. The proof is

completed.

**Theorem .**(Chung-type strong law of large numbers) *Let*{*Xn*,*n*≥}*be a sequence of*

*mean zeroψ-mixing random variables satisfying*∞_{n}_{=}*ψ*(*n*) <∞,*and let*{*an*,*n*≥}*be a*
*sequence of positive numbers satisfying* <*an*↑ ∞.*If there exists some p*∈[, ]*such that*

∞

*n*=

*E*|*Xn*|*p*
*apn*

*then*

lim

*n→∞*

*an*
*n*

*i*=

*Xi*= *a.s.* (.)

*Proof* It follows by (.) that

∞

*n*=

Var(*X*(*an*)

*n* )

*a*

*n*

≤

∞

*n*=

*E*(*X*(*an*)

*n* )
*a*

*n*

=

∞

*n*=

*EX*

*nI*(|*Xn*| ≤*an*)
*a*

*n*

≤

∞

*n*=

*E*|*Xn*|*p*
*apn*

<∞.

Therefore, we have by Theorem . that

∞

*n*=

*X*(*an*)

*n* –*EXn*(*an*)

*an* converges a.s. (.)

Since*p*∈[, ], it follows by*EXn*= that

∞

*n*=

|*EX*(*an*)

*n* |

*an*

=

∞

*n*=

|*EXnI*(|*Xn*| ≤*an*)|
*an*

=

∞

*n*=

|*EXnI*(|*Xn*|>*an*)|

*an*

≤

∞

*n*=

*E*|*Xn*|*p*
*apn*

<∞,

which implies that

∞

*n*=

*EX*(*an*)

*n*

*an* converges. (.)

Together with (.) and (.), we can see that

∞

*n*=

*X*(*an*)

*n*

*an* converges a.s. (.)

By Markov’s inequality and (.), we have

∞

*n*=

*PXn*=*Xn*(*an*)

=

∞

*n*=

*P*|*Xn*|>*an*

≤

∞

*n*=

*E*|*Xn*|*p*
*apn*

<∞. (.)

Hence, the desired result (.) follows from (.), (.), Borel-Cantelli lemma and

**3 Strong stability for weighted sums of**

**ψ**

**ψ**

**-mixing random variables**

In the previous section, we were able to get the Khintchine-Kolmogorov-type convergence
theorem for*ψ*-mixing random variables. In this section, we will study the strong stability
for weighted sums of*ψ*-mixing random variables by using the
Khintchine-Kolmogorov-type convergence theorem.

The concept of strong stability is as follows.

**Deﬁnition .** A sequence{*Yn*,*n*≥}is said to be strongly stable if there exist two
con-stant sequences{*bn*,*n*≥}and{*dn*,*n*≥}with <*bn*↑ ∞such that

*b*–* _{n}Yn*–

*dn*→ a.s.

For the deﬁnition of strong stability, one can refer to Chow and Teicher []. Many
au-thors 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 [], Bai and Cheng [], Gan and Chen [],
Kuczmaszewska [], Wu [–], Sung [], Wang*et al.*[–], Zhou [], Shen [],
Shen*et al.*[], and so on.

Our main results are as follows.

**Theorem .** *Let*{*an*,*n*≥}*and*{*bn*,*n*≥}*be two sequences of positive numbers with*
*cn*=*bn*/*anand bn*↑ ∞.*Let*{*Xn*,*n*≥}*be a sequence ofψ-mixing random variables*,*which*
*is stochastically dominated by a random variable X*.*Assume that*∞_{n}_{=}*ψ*(*n*) <∞.*Denote*
*N*(*x*) =Card{*n*:*cn*≤*x*},*x*> , ≤*p*≤.*If the following conditions are satisﬁed*

(i) *EN*(|*X*|) <∞,
(ii) _{}∞*tp*–_{P}_{(|}_{X}_{|}_{>}_{t}_{)(}∞

*t* *N*(*y*)/*yp*+*dy*)*dt*<∞,
*then there exist dn*∈**R**,*n*= , , . . . ,*such that*

*b*–_{n}*n*

*i*=

*aiXi*–*dn*→ *a.s.* (.)

*Proof* Let*Sn*=*n _{i}*

_{=}

*aiXi*,

*Tn*=

*n*

_{i}_{=}

*aiX*(

*ci*)

*i* . By Deﬁnition . and (i), we can see that

∞

*i*=

*PXi*=*Xi*(*ci*)

=

∞

*i*=

*P*|*Xi*|>*ci*

≤*C*

∞

*i*=

*P*|*X*|>*ci*

≤*CEN*|*X*|<∞. (.)

By Borel-Cantelli lemma, for any sequence{*dn*,*n*≥} ⊂**R**, the sequences{*b*–

*nTn*–*dn*}

and {*b*–

*nSn*–*dn*} converge on the same set and to the same limit. We will show that
*b*–

*n*

*n*

*i*=*ai*(*X*(*ici*)–*EX*

(*ci*)

*i* )→ a.s., which gives the theorem with*dn*=*b*–*n*

*n*

*i*=*aiEX*(*ici*).

Note that{*ai*(*X*(*ci*)

*i* –*EX*

(*ci*)

*i* ),*i*≥}is a sequence of mean zero*ψ*-mixing random variables.

It follows from*Cr*inequality, Jensen’s inequality and Lemma . that

∞

*n*=

*E*|*an*(*X*(*cn*)

*n* –*EXn*(*cn*))|*p*
*bpn*

≤*C*

∞

*n*=

*c*–* _{n}pE*|

*Xn*|

*pI*

|*Xn*| ≤*cn*

≤*C*
∞

*n*=

*c*–* _{n}pcp_{n}P*|

*X*|>

*cn*+

*E*|

*X*|

*pI*|

*X*| ≤

*cn*

≤*C*

∞

*n*=

*P*|*X*|>*cn*+*C*
∞

*n*=

*c*–_{n}p

*cn*

*tp*–*P*|*X*|>*tdt* (.)

and

∞

*n*=

*c*–_{n}p

*cn*

*tp*–*P*|*X*|>*tdt*≤

∞

*tp*–*P*|*X*|>*t*
*n*:*cn*≥t

*c*–_{n}pdt

≤*C*

∞

*tp*–*P*|*X*|>*t*
∞

*t*

*N*(*y*)/*yp*+*dy*

*dt*. (.)

The last inequality above follows from the fact that

*n*:*cn*≥t

*c*–* _{n}p*= lim

*u→∞*

*n*:*t≤cn*≤u

*c*–_{n}p

= lim

*u→∞*

*u*

*t*

*y*–*pdN*(*y*)

= lim

*u→∞*

*u*–*pN*(*u*) –*t*–*pN*(*t*) +*p*

*u*

*t*

*y*–(*p*+)*N*(*y*)*dy*

and

*u*–*pN*(*u*)≤*p*

∞

*u*

*y*–(*p*+)*N*(*y*)*dy*→ as*u*→ ∞.

Obviously,

∞

*n*=

*P*|*X*|>*cn*

≤*EN*|*X*|<∞. (.)

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

∞

*n*=

*E*|*an*(*X*(*cn*)

*n* –*EXn*(*cn*))|*p*
*bpn*

<∞. (.)

Therefore,

*b*–_{n}*n*

*i*=

*aiX*(*ci*)

*i* –*EX*

(*ci*)

*i*

→ a.s.,

following from (.), Theorem . and Kronecker’s lemma immediately. The desired result

is obtained.

**Corollary .** *Let the conditions of Theorem*.*be satisﬁed*,*and let EXn*= *for n*≥.
*Assume that*_{}∞*EN*(|*X*|/*s*)*ds*<∞.*Then b*–

*n*

*n*

*i*=*aiXi*→*a*.*s*.

*Proof* By Theorem ., we only need to prove that

*b*–_{n}*n*

*i*=

*aiEX*(*ci*)

In fact,

∞

*i*=

*ai*|*EXi*(*ci*)|
*bi* =

∞

*i*=

*c*–_{i}*EXiI*|*Xi*| ≤*ci*≤
∞

*i*=

*c*–_{i}*E*|*Xi*|*I*|*Xi*|>*ci*

≤

∞

*i*=

*c*–_{i}

*ciP*

|*Xi*|>*ci*

+

∞

*ci*

*P*|*Xi*|>*t*

*dt*

≤*CEN*|*X*|+*C*

∞

*EN*|*X*|/*sds*<∞,

which implies (.) by Kronecker’s lemma. We complete the proof of the corollary.

**Theorem .** *Let*{*an*,*n*≥}*and*{*bn*,*n*≥}*be two sequences of positive numbers with cn*=
*bn*/*anand bn*↑ ∞.*Let*{*Xn*,*n*≥}*be a sequence of mean zeroψ-mixing random variables*,
*which is stochastically dominated by a random variable X*.*Assume that*∞_{n}_{=}*ψ*(*n*) <∞.

*Denote N*(*x*) =Card{*n*:*cn*≤*x*},*x*> , ≤*p*≤.*If the following conditions are satisﬁed*
(i) *EN*(|*X*|) <∞,

(ii) _{}∞*EN*(|*X*|/*s*)*ds*<∞,
(iii) max≤j≤n*cpj*

∞

*i*=*nc*

–*p*
*i* =*O*(*n*),
*then*

*b*–_{n}*n*

*i*=

*aiXi*→ *a.s.* (.)

*Proof* By condition (i) and (.), we only need to prove that*b*–

*n*

*n*

*i*=*aiXi*(*ci*)→ a.s. For

this purpose, it suﬃces to show that

*b*–_{n}*n*

*i*=

*ai*

*X*(*ci*)

*i* –*EX*

(*ci*)

*i*

→ a.s. (.)

and

*b*–_{n}*n*

*i*=

*aiEX*(*ci*)

*i* → as*n*→ ∞. (.)

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

To prove (.), we set*ε*= and*εn*=max≤j≤n*cj*for*n*≥. It follows from*Cr*inequality,

Jensen’s inequality and Lemma . that

∞

*n*=

*E*|*an*(*X*(*cn*)

*n* –*EXn*(*cn*))|*p*
*bpn*

≤*C*

∞

*n*=

*c*–* _{n}pE*|

*Xn*|

*pI*|

*Xn*| ≤

*cn*

≤*C*

∞

*n*=

*P*|*X*|>*cn*

+*C*

∞

*n*=

*c*–* _{n}pE*|

*X*|

*pI*|

*X*| ≤

*cn*

.

Obviously,

∞

*n*=

*P*|*X*|>*cn*

and

∞

*n*=

*c*–* _{n}pE*|

*X*|

*pI*|

*X*| ≤

*cn*

≤

∞

*n*=

*c*–* _{n}pE*|

*X*|

*pI*|

*X*| ≤

*εn*

≤

∞

*j*=

*ε _{j}pPεj*–<|

*X*| ≤

*εj*

_{}∞

*n*=*j*

*c*–* _{n}p*≤

*C*∞

*j*=

*P*|*X*|>*εj*–

≤*C*

+

∞

*n*=

*P*|*X*|>*cn*

≤*C* +*EN*|*X*|<∞.

Therefore,

∞

*n*=

*E*|*an*(*X*(*cn*)

*n* –*EXn*(*cn*))|*p*
*bpn*

<∞, (.)

following from the statements above. By Theorem . and Kronecker’s lemma, we can

obtain (.) immediately. The proof is completed.

**Theorem .** *Let*{*an*,*n*≥}*and*{*bn*,*n*≥}*be two sequences of positive numbers with*
*cn*=*bn*/*anand bn*↑ ∞.*Let*{*Xn*,*n*≥}*be a sequence ofψ-mixing random variables*,*which*
*is stochastically dominated by a random variable X*.*Assume that*∞_{n}_{=}*ψ*(*n*) <∞.*Deﬁne*
*N*(*x*) =Card{*n*:*cn*≤*x*},*R*(*x*) =* _{x}*∞

*N*(

*y*)

*y*–

*dy*,

*x*> .

*If the following conditions are satisﬁed*

(i) *N*(*x*) <∞*for anyx*> ,
(ii) *R*() =_{}∞*N*(*y*)*y*–_{dy}_{<}_{∞}_{,}

(iii) *EX**R*(|*X*|) <∞,

*then there exist dn*∈**R**,*n*= , , . . . ,*such that*

*b*–_{n}*n*

*i*=

*aiXi*–*dn*→ *a.s.* (.)

*Proof* Since*N*(*x*) is nondecreasing, then for any*x*>

*R*(*x*)≥*N*(*x*)

∞

*x*

*y*–*dy*=
*x*

–_{N}_{(}_{x}_{),} _{(.)}

which implies that*EN*(|*X*|)≤*EX*_{R}_{(|}_{X}_{|) <}_{∞. Therefore,}

∞

*i*=

*PXi*=*X*(*ci*)

*i*

=

∞

*i*=

*P*|*Xi*|>*ci*

≤*C*

∞

*i*=

*P*|*X*|>*ci*≤*CEN*|*X*|<∞. (.)

By Borel-Cantelli lemma for any sequence{*dn*,*n*≥} ⊂**R**,{*b*–* _{n}Sn*–

*dn*}and{

*b*–

*–*

_{n}Tn*dn*}

converge on the same set and to the same limit. We will show that *b*–

*n*

*n*

*i*=*ai*(*X*
(*ci*)

*i* –

*EX*(*ci*)

*i* )→ a.s., which gives the theorem with *dn*=*b*–*n*

*n*

*i*=*aiEX*
(*ci*)

Lemma . that

∞

*n*=

Var(*anX*(*cn*)

*n* )

*b*

*n*

≤

∞

*n*=

*c*–_{n}*EX*(*cn*)

*n*

=

∞

*n*=

*c*–_{n}*EX _{n}*

*I*|

*Xn*| ≤

*cn*

≤*CEN*|*X*|+*C*
∞

*n*=

*c*–_{n}*EX**I*|*X*| ≤*cn*

(.)

and

∞

*n*=

*c*–_{n}*EX**I*|*X*| ≤*cn*=

*n*:*cn*≤

*c*–_{n}*EX**I*|*X*| ≤*cn*+

*n*:*cn*>

*c*–_{n}*EX**I*|*X*| ≤*cn*

.

=*I*+*I*. (.)

Since*N*() =Card{*n*:*cn*≤} ≤*R*() <∞from (.) and (ii), it follows that*I*<∞. For

*I*, we have

*I*=

*n*:*cn*>

*c*–_{n}*EX**I*|*X*| ≤*cn*

=

∞

*k*=

*k*–<*cn*≤k

*c*–_{n}*EX**I*|*X*| ≤*cn*

≤

∞

*k*=

*N*(*k*) –*N*(*k*– )(*k*– )–*EX**I*|*X*| ≤

+

∞

*k*=

*N*(*k*) –*N*(*k*– )(*k*– )–*EX**I* <|*X*| ≤*k*

.

=*I*+*I*,

*I*≤*C*

∞

*k*=

*N*(*k*) –*N*(*k*– )

∞

*j*=*k*–

*j*–=*C*
∞

*j*=

*j*–
*j*+

*k*=

*N*(*k*) –*N*(*k*– )

≤*C*

∞

*j*=

(*j*+ )–*N*(*j*+ )≤*C*

∞

*y*–*N*(*y*)*dy*<∞.

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

*I*≤

∞

*m*=

*EX**Im*– <|*X*| ≤*m*
∞

*k*=*m*

*N*(*k*)(*k*– )––*k*–

≤*C*

∞

*m*=

*EX**Im*– <|*X*| ≤*m*
∞

*k*=*m*

*k*+

*k*

*N*(*x*)*x*–*dx*

≤*C*

∞

*m*=

*EX**R*|*X*|*Im*– <|*X*| ≤*m*≤*CEX**R*|*X*|<∞.

Therefore,

∞

*n*=

Var(*anX*(*cn*)

*n* )

*b*

*n*

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

*b*–_{n}*n*

*i*=

*aiX*(*ci*)

*i* –*EX*

(*ci*)

*i*

→ a.s. (.)

Taking*dn*=*b*–*n*

*n*

*i*=*aiEXi*(*ci*), we have*b*–*n*

*n*

*i*=*aiX*(*ici*)–*dn*→ a.s. The proof is

com-pleted.

**Corollary .** *Let the conditions of Theorem* . *be satisﬁed*. *If EXn*= , *n*≥ *and*

∞

*EN*(|*X*|/*s*)*ds*<∞,*then b*–*n*

*n*

*i*=*aiXi*→*a*.*s*.

In the following, we denote*α*(*x*) :**R**+→**R**+ as a positive and nonincreasing function

with*an*=*α*(*n*),*bn*=*n _{i}*

_{=}

*ai*,

*cn*=

*bn*/

*an*,

*n*≥, where

<*bn*↑ ∞, (.)

<lim inf

*n→∞* *n*

–_{c}

*nα*(log*cn*)≤lim sup

*n→∞* *n*

–_{c}

*nα*(log*cn*) <∞, (.)

*xα*log+*x* is nonincreasing for*x*> . (.)

**Theorem .** *Let*{*Xn*,*n*≥}*be a sequence of identically distributedψ-mixing random*
*variables with*∞_{n}_{=}*ψ*(*n*) <∞.*If E*|*X*|*α*(log+|*X*|) <∞,*then there exist dn*∈**R**,*n*= , , . . . ,

*such that b*–

*n*

*n*

*i*=*aiXi*–*dn*→*a*.*s*.

*Proof* Since*α*(*x*) is positive and nonincreasing for*x*> and <*bn*↑ ∞, it follows that
*cn*↑ ∞. By (.), we can choose constants*m*∈**N**,*C*> ,*C*> such that for*n*≥*m*,

*C**n*≤*cnα*(log*cn*)≤*C**n*. (.)

Therefore, for*n*≥*m*, we have * _{c}*

*n* ≤

*α*(log*cm*)

*C**n* , which implies that

∞

*j*=*m*
*c*–* _{j}* ≤

∞

*j*=*m*

*α*(log*cm*)

*C**j*

≤*α*(log*cm*)

*C**m*

. (.)

By (.)-(.), it follows that

∞

*j*=*m*

*E*(*ajX _{j}*(

*cj*))

*b*

*j*

≤

∞

*j*=*m*
*c*–_{j}

{|X|≤c*m–*}

*X*_{}*dP*+

*j*

*i*=*m*

{c*i–*<|X|≤c*i*}

*X*_{}*dP*

≤*C*+

∞

*j*=*m*
*c*–_{j}

*j*

*i*=*m*

{c*i–*<|X|≤c*i*}

*X*_{}*dP*

≤*C*+*C*

∞

*i*=*m*

*i*–*α*(log*ci*)

{c*i–*<|X|≤c*i*}

*X*_{}*dP*

≤*C*+*C*

∞

*i*=*m*

*α*(log*ci*)

{c*i–*<|X|≤c*i*}

|*X*|*dP*

≤*C*+*C*

∞

*i*=*m*

{c*i–*<|X|≤c*i*}

|*X*|*α*

log+|*X*|

Therefore,

∞

*j*=

Var(*ajX _{j}*(

*cj*))

*b*

*j*

≤

∞

*j*=

*E*(*ajX _{j}*(

*cj*))

*b*

*j*

<∞, (.)

which implies that

*b*–_{n}*n*

*i*=

*aiX*(*ci*)

*i* –*EX*

(*ci*)

*i*

→ a.s. (.)

from Theorem . and Kronecker’s lemma. By (.) and (.) again, we have

∞

*j*=*m*

*P*|*Xj*|>*cj*

≤

∞

*j*=*m*
*P*|*Xj*|*α*

log+|*Xj*|

≥*cjα*(log*cj*)

≤

∞

*j*=*m*
*P*|*X*|*α*

log+|*X*|

≥*C**j*

<∞,

∞

*j*=

*PXj*=*X _{j}*(

*cj*)=

∞

*j*=

*P*|*Xj*|>*cj*=

*m*–

*j*=

*P*|*Xj*|>*cj*+

∞

*j*=*m*

*P*|*Xj*|>*cj*<∞.

By Borel-Cantelli lemma, we have*P*(*Xj*=*X*(* _{j}cj*), i.o.) = . Together with (.), we can see
that

*b*–_{n}*n*

*i*=

*aiXi*–*EX*(*ci*)

*i*

→ a.s. (.)

Taking*dn*=*b*–_{n}*n _{i}*

_{=}

*aiEX*(

*ci*)

*i* for*n*≥, we get the desired result.

**Theorem .** *Let* {*Xn*,*n* ≥ } *be a sequence of* *ψ-mixing random variables with*

∞

*n*=*ψ*(*n*) <∞.*If for some*≤*p*≤,

∞

*n*=

*n*–*pEXnα*log+|*Xn*|*p*<∞,

*then there exist dn*∈**R**,*n*= , , . . . ,*such that b*–*n*

*n*

*i*=*aiXi*–*dn*→*a*.*s*.
*Proof* Similar to the proof of Theorem ., it is easily seen that

∞

*j*=

*PXj*=*X _{j}*(

*cj*)≤

*m*– +

∞

*j*=*m*

*P*|*Xj*|*α*log+|*Xj*|≥*cjα*(log*cj*)

≤*m*– +

∞

*j*=*m*

*P*|*Xj*|*α*log+|*Xj*|≥*C**j*

<∞.

By Borel-Cantelli lemma for any sequence{*dn*,*n*≥} ⊂**R**, the sequences{*b*–_{n}*n _{i}*

_{=}

*aiXi*–

*dn*}and{*b*–

*n*

*n*

*i*=*aiX*
(*ci*)

that*b*–

*n*

*n*

*i*=*ai*(*X*
(*ci*)

*i* –*EX*

(*ci*)

*i* )→ a.s., which gives the theorem with*dn*=*b*–*n*

*n*

*i*=*aiEX*
(*ci*)

*i* .

Note that{*ai*(*X*(*ci*)

*i* –*EX*

(*ci*)

*i* )/*bi*,*i*≥}is a sequence of mean zero*ψ*-mixing random

vari-ables. By*Cr*inequality and Jensen’s inequality, we can see that

∞

*j*=

*E*|*aj*(*X _{j}*(

*cj*)–

*EX*(

*))|*

_{j}cj*p*

*bp _{j}* ≤

*C*(

*m*– ) +

*C*∞

*j*=*m*

*c*–* _{j}pE*|

*Xj*|

*pI*|

*Xj*| ≤

*cj*

≤*C*(*m*– ) +*C*
∞

*j*=*m*

*j*–*pα*(log*cj*)*pE*|*Xj*|*pI*|*Xj*| ≤*cj*

≤*C*(*m*– ) +*C*
∞

*j*=

*j*–*pEXjα*log+|*Xj*|*p*<∞.

It follows by Theorem . that*b*–

*n*

*n*

*i*=*ai*(*Xi*(*ci*)–*EX*

(*ci*)

*i* )→ a.s. The proof is completed.

**Corollary .** *Let the conditions of Theorem*.*be satisﬁed*.*Furthermore*,*suppose that*
*EXn*= *and*∞_{n}_{=}_{}∞*P*(|*Xn*|>*scn*)*ds*<∞,*then b*–

*n*

*n*

*i*=*aiXi*→*a*.*s*.

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

All authors read and approved the ﬁnal manuscript.

**Author details**

1_{School of Mathematical Science, 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 grateful to the editor Andrei Volodin and an anonymous referee for careful reading of the manuscript and valuable suggestions, which helped in improving an earlier version of this paper. This work was supported by the Natural Science Project of Department of Education of Anhui Province (KJ2011z056) and the National Natural Science Foundation of China (11201001).

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

1. Blum, JR, Hanson, DL, Koopmans, L: On the strong law of large numbers for a class of stochastic process. Z. Wahrscheinlichkeitstheor. Verw. Geb.2, 1-11 (1963)

2. Yang, SC: Almost sure convergence of weighted sums of mixing sequences. J. Syst. Sci. Math. Sci.15(3), 254-265 (1995)

3. Wu, QY: Strong consistency of*M*estimator in linear model for*ρ*-mixing,*ϕ*-mixing,*ψ*-mixing samples. Math. Appl.

17(3), 393-397 (2004)

4. Wang, XJ, Hu, SH, Shen, Y, Yang, WZ: Maximal inequality for*ψ*-mixing sequences and its applications. Appl. Math.
Lett.23, 1156-1161 (2010)

5. Zhu, YC, Deng, X, Pan, J, Ling, JM, Wang, XJ: Strong law of large numbers for sequences of*ψ*-mixing random
variables. J. Math. Study45(4), 404-410 (2012)

6. Pan, J, Zhu, YC, Zou, WY, Wang, XJ: Some convergence results for sequences of*ψ*-mixing random variables. Chin.
Q. J. Math.28(1), 111-117 (2013)

7. Jamison, B, Orey, S, Pruitt, W: Convergence of weighted averages of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.4, 40-44 (1965)

8. Wang, XJ, Hu, SH, Yang, WZ, Wang, XH: On complete convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables. Abstr. Appl. Anal.2012, Article ID 315138 (2012) 9. Tang, XF: Some strong laws of large numbers for weighted sums of asymptotically almost negatively associated

random variables. J. Inequal. Appl.2013, 4 (2013)

10. Shen, AT, Wu, RC: Strong and weak convergence for asymptotically almost negatively associated random variables. Discrete Dyn. Nat. Soc.2013, Article ID 235012 (2013)

12. Hu, TC, Taylor, RL: On the strong law for arrays and for the bootstrap mean and variance. Int. J. Math. Math. Sci.20(2), 375-382 (1997)

13. Bai, ZD, Cheng, PE: Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett.46, 105-112 (2000)

14. Gan, SX, Chen, PY: On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables. Acta Math. Sci., Ser. B27(2), 283-290 (2007)

15. Kuczmaszewska, A: On Chung-Teicher type strong law of large numbers for*ρ*∗-mixing random variables. Discrete
Dyn. Nat. Soc.2008, Article ID 140548 (2008)

16. Wu, QY: Complete convergence for negatively dependent sequences of random variables. J. Inequal. Appl.2010, Article ID 507293 (2010)

17. Wu, QY: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl.2010, Article ID 383805 (2010)

18. Wu, QY: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl.2012, 50 (2012)

19. Sung, SH: On the strong convergence for weighted sums of random variables. Stat. Pap.52, 447-454 (2011) 20. Wang, XJ, Li, XQ, Hu, SH, Yang, WZ: Strong limit theorems for weighted sums of negatively associated random

variables. Stoch. Anal. Appl.29, 1-14 (2011)

21. Wang, XJ, Hu, SH, Yang, WZ: Complete convergence for arrays of rowwise asymptotically almost negatively associated random variables. Discrete Dyn. Nat. Soc.2011, Article ID 717126 (2011)

22. Wang, XJ, Hu, SH, Volodin, AI: Strong limit theorems for weighted sums of NOD sequence and exponential inequalities. Bull. Korean Math. Soc.48(5), 923-938 (2011)

23. Wang, XJ, Li, XQ, Yang, WZ, Hu, SH: On complete convergence for arrays of rowwise weakly dependent random variables. Appl. Math. Lett.25, 1916-1920 (2012)

24. Wang, XJ, Hu, SH, Yang, WZ: Complete convergence for arrays of rowwise negatively orthant dependent random variables. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.106, 235-245 (2012)

25. Zhou, XC, Tan, CC, Lin, JG: On the strong laws for weighted sums of*ρ*∗-mixing random variables. J. Inequal. Appl.

2011, Article ID 157816 (2011)

26. Shen, AT: Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables. J. Inequal. Appl.2011, 93 (2011)

27. Shen, AT, Wu, RC, Chen, Y, Zhou, Y: Complete convergence of the maximum partial sums for arrays of rowwise of AANA random variables. Discrete Dyn. Nat. Soc.2013, Article ID 741901 (2013)

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