R E S E A R C H
Open Access
Strong convergence properties for
ψ
-mixing
random variables
Huai Xu
1and Ling Tang
2**Correspondence:
2Department 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 forψ
-mixing random variables. In addition, the strong stability for weighted sums ofψ
-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 variables1 Introduction
Let (,F,P) be a fixed probability space. The random variables we deal with are all defined on (,F,P). Throughout the paper, letI(A) be the indicator function of the setA. For random variableX, denoteX(c)=XI(|X| ≤c) for somec> . Denotelog+x=. ln max(e,x).
Candcdenote 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 letSn=ni=Xifor eachn≥. Letnandmbe positive integers. WriteFm
n =
σ(Xi,n≤i≤m). Givenσ-algebrasB,RinF, 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≥
ψFk,Fk∞+n, n≥.
Definition . A sequence{Xn,n≥} of random variables is said to be a sequence of
ψ-mixing random variables ifψ(n)↓ asn→ ∞.
The concept ofψ-mixing random variables was introduced by Blumet al.[] and some applications have been found. See, for example, Blumet al.[] for strong law of large num-bers, Yang [] for almost sure convergence of weighted sums, Wu [] for strong consistency ofMestimator in linear model, Wanget al.[] for maximal inequality and Hájek-Rényi-type inequality, strong growth rate and the integrability of the supremum, Zhuet al.[] for
strong convergence properties, Panet 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 forψ-mixing random variables. In addition, we will study the strong stability for weighted sums ofψ-mixing random variables, which generalizes the corresponding one of independent random variables.
For independent and identically distributed random variable sequences, Jamisonet 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=ni=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)/ydy)dF(x) <∞, then
Wn– 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 ofψ-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 forψ-mixing random variables.
Lemma .(cf.Wanget 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+
EXi 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=
EXi q/
(.)
for every n≥.
Definition . A sequence{Xn,n≥} of random variables is said to be stochastically dominated by a random variableXif there exists a constantCsuch that
P|Xn|>x
≤CP|X|>x (.)
for allx≥ andn≥.
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 Wanget 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 forψ-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 thatEXn= for alln≥. For anyε> , 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 forψ-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=
EXn(c) converges, (.)
∞
n=
VarXn(c)<∞, (.)
then∞n=Xnconverges almost surely.
Proof According to (.) and Theorem ., we have
∞
n=
Xn(c)–EX(nc) converges a.s. (.)
It follows by (.) and (.) that
∞
n=
X(nc) converges a.s. (.)
Obviously, (.) implies that
∞
n=
PXn=Xn(c)=
∞
n=
P|Xn|>c<∞. (.)
It follows by (.) and Borel-Cantelli lemma that
PXn=Xn(c), i.o.= . (.)
Finally, combining (.) with (.), we can get that∞n=Xnconverges 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. (.)
Sincep∈[, ], it follows byEXn= 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
-mixing random variablesIn the previous section, we were able to get the Khintchine-Kolmogorov-type convergence theorem forψ-mixing random variables. In this section, we will study the strong stability for weighted sums ofψ-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–nYn–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 [], Wanget al.[–], Zhou [], Shen [], Shenet 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 LetSn=ni=aiXi,Tn=ni=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 withdn=b–n
n
i=aiEX(ici).
Note that{ai(X(ci)
i –EX
(ci)
i ),i≥}is a sequence of mean zeroψ-mixing random variables.
It follows fromCrinequality, Jensen’s inequality and Lemma . that
∞
n=
E|an(X(cn)
n –EXn(cn))|p bpn
≤C
∞
n=
c–npE|Xn|pI
|Xn| ≤cn
≤C ∞
n=
c–npcpnP|X|>cn+E|X|pI|X| ≤cn
≤C
∞
n=
P|X|>cn+C ∞
n=
c–np
cn
tp–P|X|>tdt (.)
and
∞
n=
c–np
cn
tp–P|X|>tdt≤
∞
tp–P|X|>t n:cn≥t
c–npdt
≤C
∞
tp–P|X|>t ∞
t
N(y)/yp+dy
dt. (.)
The last inequality above follows from the fact that
n:cn≥t
c–np= lim
u→∞
n:t≤cn≤u
c–np
= 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→ asu→ ∞.
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≤ncpj
∞
i=nc
–p i =O(n), then
b–n n
i=
aiXi→ a.s. (.)
Proof By condition (i) and (.), we only need to prove thatb–
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 → asn→ ∞. (.)
Equation (.) follows from the proof of Corollary . immediately.
To prove (.), we setε= andεn=max≤j≤ncjforn≥. It follows fromCrinequality,
Jensen’s inequality and Lemma . that
∞
n=
E|an(X(cn)
n –EXn(cn))|p bpn
≤C
∞
n=
c–npE|Xn|pI|Xn| ≤cn
≤C
∞
n=
P|X|>cn
+C
∞
n=
c–npE|X|pI|X| ≤cn
.
Obviously,
∞
n=
P|X|>cn
and
∞
n=
c–npE|X|pI|X| ≤cn
≤
∞
n=
c–npE|X|pI|X| ≤εn
≤
∞
j=
εjpPεj–<|X| ≤εj
∞
n=j
c–np≤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) EXR(|X|) <∞,
then there exist dn∈R,n= , , . . . ,such that
b–n n
i=
aiXi–dn→ a.s. (.)
Proof SinceN(x) is nondecreasing, then for anyx>
R(x)≥N(x)
∞
x
y–dy= x
–N(x), (.)
which implies thatEN(|X|)≤EXR(|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–nSn–dn}and{b–nTn–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 EXnI|Xn| ≤cn
≤CEN|X|+C ∞
n=
c–n EXI|X| ≤cn
(.)
and
∞
n=
c–n EXI|X| ≤cn=
n:cn≤
c–n EXI|X| ≤cn+
n:cn>
c–n EXI|X| ≤cn
.
=I+I. (.)
SinceN() =Card{n:cn≤} ≤R() <∞from (.) and (ii), it follows thatI<∞. For
I, we have
I=
n:cn>
c–n EXI|X| ≤cn
=
∞
k=
k–<cn≤k
c–n EXI|X| ≤cn
≤
∞
k=
N(k) –N(k– )(k– )–EXI|X| ≤
+
∞
k=
N(k) –N(k– )(k– )–EXI <|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<∞.
SinceN(x) is nondecreasing andR(x) is nonincreasing, we have
I≤
∞
m=
EXIm– <|X| ≤m ∞
k=m
N(k)(k– )––k–
≤C
∞
m=
EXIm– <|X| ≤m ∞
k=m
k+
k
N(x)x–dx
≤C
∞
m=
EXR|X|Im– <|X| ≤m≤CEXR|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. (.)
Takingdn=b–n
n
i=aiEXi(ci), we haveb–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 denoteα(x) :R+→R+ as a positive and nonincreasing function
withan=α(n),bn=ni=ai,cn=bn/an,n≥, where
<bn↑ ∞, (.)
<lim inf
n→∞ n
–c
nα(logcn)≤lim sup
n→∞ n
–c
nα(logcn) <∞, (.)
xαlog+x is nonincreasing forx> . (.)
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 Sinceα(x) is positive and nonincreasing forx> and <bn↑ ∞, it follows that cn↑ ∞. By (.), we can choose constantsm∈N,C> ,C> such that forn≥m,
Cn≤cnα(logcn)≤Cn. (.)
Therefore, forn≥m, we have c
n ≤
α(logcm)
Cn , which implies that
∞
j=m c–j ≤
∞
j=m
α(logcm)
Cj
≤α(logcm)
Cm
. (.)
By (.)-(.), it follows that
∞
j=m
E(ajXj(cj))
b
j
≤
∞
j=m c–j
{|X|≤cm–}
XdP+
j
i=m
{ci–<|X|≤ci}
XdP
≤C+
∞
j=m c–j
j
i=m
{ci–<|X|≤ci}
XdP
≤C+C
∞
i=m
i–α(logci)
{ci–<|X|≤ci}
XdP
≤C+C
∞
i=m
α(logci)
{ci–<|X|≤ci}
|X|dP
≤C+C
∞
i=m
{ci–<|X|≤ci}
|X|α
log+|X|
Therefore,
∞
j=
Var(ajXj(cj))
b
j
≤
∞
j=
E(ajXj(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(logcj)
≤
∞
j=m P|X|α
log+|X|
≥Cj
<∞,
∞
j=
PXj=Xj(cj)=
∞
j=
P|Xj|>cj=
m–
j=
P|Xj|>cj+
∞
j=m
P|Xj|>cj<∞.
By Borel-Cantelli lemma, we haveP(Xj=X(jcj), i.o.) = . Together with (.), we can see that
b–n n
i=
aiXi–EX(ci)
i
→ a.s. (.)
Takingdn=b–n ni=aiEX(ci)
i forn≥, 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–pEXnαlog+|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=Xj(cj)≤m– +
∞
j=m
P|Xj|αlog+|Xj|≥cjα(logcj)
≤m– +
∞
j=m
P|Xj|αlog+|Xj|≥Cj
<∞.
By Borel-Cantelli lemma for any sequence{dn,n≥} ⊂R, the sequences{b–n ni=aiXi–
dn}and{b–
n
n
i=aiX (ci)
thatb–
n
n
i=ai(X (ci)
i –EX
(ci)
i )→ a.s., which gives the theorem withdn=b–n
n
i=aiEX (ci)
i .
Note that{ai(X(ci)
i –EX
(ci)
i )/bi,i≥}is a sequence of mean zeroψ-mixing random
vari-ables. ByCrinequality and Jensen’s inequality, we can see that
∞
j=
E|aj(Xj(cj)–EX(jcj))|p
bpj ≤C(m– ) +C ∞
j=m
c–jpE|Xj|pI|Xj| ≤cj
≤C(m– ) +C ∞
j=m
j–pα(logcj)pE|Xj|pI|Xj| ≤cj
≤C(m– ) +C ∞
j=
j–pEXjαlog+|Xj|p<∞.
It follows by Theorem . thatb–
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
1School of Mathematical Science, Anhui University, Hefei, 230601, P.R. China.2Department 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
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doi:10.1186/1029-242X-2013-360
