# On Chung-Teicher Type Strong Law of Large Numbers forρ∗-Mixing Random Variables

Authors: Anna Kuczmaszewska
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for independent random variables were generalized on the case ofρ∗-mixing sequence. The main result was applied to obtain a Marcinkiewicz SLLN.


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## Body

## 1. Introduction

Let{Xn,n≥1} be a sequence of random variables defined on
the probability space (Ω,ℱ,P) with value in a real space ℝ and let Sn=∑i=1nXi.
We say that the sequence {Xn,n≥1} satisfies the strong law of large numbers
(SLLN) if there exists some increasing sequence {bn,n≥1} and some sequence {an,n≥1} such that(1.1)∑i=1n(Xi−ai)bn⟶0a.s.asn⟶∞.In this paper, we consider the
strong law of large numbers for sequences of dependent random variables which
are said to be ρ*-mixing. To introduce the concept of ρ*-mixing sequence, we need the maximal
correlation coefficient defined as follows:(1.2)ρ*(k)=sup⁡S,T(sup⁡X∈L2(ℱS),Y∈L2(ℱT)cov(X,Y)VarX⁢⋅VarY),where S, T are the finite subsets of positive integers
such that dist (S,T)=infx∈S,y∈T|x−y|≥k and ℱW is the σ-field generated by the random variables {Xi,i∈W⊂ℕ}.Definition 1.1.
A sequence of random variables{Xn,n≥1} is said to be a ρ*-mixing sequence if (1.3)lim⁡n→∞⁢ρ*(n)<1.ρ*-mixing random variables were investigated by
many authors. Various moment inequalities for sums and maximum of partial sums
can be found in papers by Bradley [1], Bryc and Smoleński [2], Peligrad [3], Peligrad and Gut [4],
and Utev and Peligrad [5]. These inequalities are used in many papers concerning
the problems of invariance principle (Utev and Peligrad [5]), CLT (Peligrad
[3]), or complete convergence for some stochastically dominated sequence of ρ*-mixing random variables (Cai [6]), and for an
array of rowwise ρ*-mixing random variables (Zhu [7]). They will
be also important in our further consideration.The aim of this paper is to give some sufficient
conditions for SLLN for a sequence ofρ*-mixing random variables without assumptions
of identical distribution and stochastical domination. The result presented in
this paper is obtained by using the maximal type inequality and the following
strong law of large numbers proved by Fazekas and Klesov [8].Theorem 1.2 (Fazekas and Klesov [8]).
Let{bn,n≥1} be a nondecreasing, unbounded sequence of
positive numbers. Let {αn,n≥1} be nonnegative numbers. Let r be a fixed positive number. Assume that for
each n≥1: (1.4)E[max⁡1≤i≤n⁢|Si|]r≤∑i=1nαi.If(1.5)∑i=1∞αibir<∞,then(1.6)lim⁡n→∞Snbn=0a.s.Using this theorem, we are going to show that classical Kolmogorov,
Chung, and Teicher's strong law of large numbers for independent random
variables{Xn,n≥1} (Chung [9] and Teicher [10]) can be generalized
to the case of ρ*-mixing sequences.In our further consideration, we need the following
result.Lemma 1.3 (Utev and Peligrad [5]).
Let{Xn,n≥1} be a ρ*-mixing sequence with EXn=0, E|Xn|q<∞, n≥1, q≥2.
Then, there exists a positive constant c such that (1.7)E⁢max⁡1≤k≤n⁢|∑i=1kXi|q≤c[∑i=1nE|Xi|q+(∑i=1nEXi2)q/2],∀n≥1.LetC denote a constant which is not necessary the
same in its each appearance.


## 2. The Main Result

Let{φn,n≥1} be a sequence of nonnegative, even, continuous
and nondecreasing on (0,∞) functions φn:ℝ→ℝ+ with limx→∞φn(x)=∞ and such that for all n≥1 and 1<p≤2:(2.1)(a)φn(x)/x↘or(b)φn(x)/x↗,φn(x)/xp↘asx⟶∞.Theorem 2.1.
Let{Xn,n≥1} be a sequence of ρ*-mixing random variables and let {bn,n≥1} be an increasing sequence of positive real
numbers. Let 1<p≤2.
Assume that{φn,n≥1} satisfies (a) in (2.1) with
(A)
(2.2)(A)∑i=2∞bi−pE(φip(|Xi|)φip(bi)+φip(|Xi|))∑j=1i−1bjpE(φjp(|Xj|)φjp(bj)+φjp(|Xj|))<∞,(B)
(2.3)(B)∑n=1∞P[|Xn|≥an]<∞,
for some
sequence{an,n≥1} of positive numbers such that
(C)
(2.4)(C)∑n=1∞E(φn2p[min⁡(|Xn|,an)]φn2p(bn)+φn2p(|Xn|))<∞,
or {φn,n≥1} satisfies (b) in (2.1) with
(A1)
(2.5)(A1)∑i=2∞bi−pE(φi(|Xi|)φi(bi)+φi(|Xi|))∑j=1i−1bjpE(φj(|Xj|)φj(bj)+φj(|Xj|))<∞,
and (B) for some sequence {an,n≥1} of positive numbers such that
(C1)
(2.6)(C1)∑n=1∞E(φn2[min⁡(|Xn|,an)]φn2(bn)+φn2(|Xn|))<∞.
Then,(2.7)bn−1∑i=1n{Xi−E(XiI[|Xi|<bi])}⟶0a.s.n⟶∞.Proof.
LetXi′=XiI[|Xi|<bi], Xi*=Xi′−EXi′, Sn′=∑i=1nXi*,
and Sn=∑i=1nXi.
Then,(2.8)∑n=1∞P[Xn′⁢≠Xn]=∑n=1∞P[|Xn|≥bn]≤∑n=1∞P[2φn2r(|Xn|)≥φn2r(bn)+φn2r(|Xn|)]≤C∑n=1∞{E(φn2r(|Xn|)φn2r(bn)+φn2r(|Xn|)⁢⋅I[|Xn|≥an])+E(φn2r(|Xn|)φn2r(bn)+φn2r(|Xn|)⁢⋅I[|Xn|<an])}≤C{∑n=1∞P[|Xn|≥an]+∑n=1∞E(φn2r[min⁡(|Xn|,an)]φn2r(bn)+φn2r(|Xn|))}<∞for r=p in the case (a) or r=1 in the case (b). Hence, the sequences {Xn,n≥1} and {Xn′,n≥1} are equivalent in Khinchin's sense.
Thus, we need only to show that(2.9)Sn′bn⟶0a.s.n⟶∞.By Lemma 1.3, for q=4,
we have(2.10)E[max⁡1≤i≤n⁢|Si′|]4≤C[∑i=1nEXi*4+(∑i=1nEXi*2)2]≤C(2∑i=1nEXi*4+2∑i=2nEXi*2∑j=1i−1EXj*2)≤C[EX1′4+∑i=2n(EXi′4+EXi′2∑j=1i−1EXj′2)].By Theorem 1.2 applied
with(2.11)α1=EX1′4,αi=EXi′4+EXi′2∑j=1i−1EXj′2fori=2,3,…,n,we see that in order to show
(2.9) it is enough to prove that(2.12)∑i=1∞αibi4<∞,which holds if(2.13)∑i=1∞EXi′4bi4<∞,∑i=2∞EXi′2bi4∑j=1i−1EXj′2<∞.Put I:=∑i=1∞P[|Xi|≥ai].
Then, we have(2.14)∑i=1∞EXi′4bi4≤∑i=1∞E(XiI[|Xi|<min⁡(ai,bi)])4bi4+I≤∑i=1∞E(|Xi|2pI[|Xi|<min⁡(ai,bi)])bi2p+I.Moreover, we note
that(2.15)ab⁢⋅I[a≤b]≤2⁢⋅aa+b∀a,b>0.Hence, in case (a) we get, by (B) and (C),(2.16)∑i=1∞EXi′4bi4≤∑i=1∞E(φi2p(|Xi|)φi2p(bi)I[|Xi|<min⁡(ai,bi)])+I≤2∑i=1∞E(φi2p[min⁡(|Xi|,ai)]φi2p(bi)+φi2p(|Xi|))+I<∞,while in case (b) we get, by (B) and (C1),(2.17)∑i=1∞EXi′4bi4≤2∑i=1∞E(φi2[min⁡(|Xi|,ai)]φi2(bi)+φi2(|Xi|))+I<∞.Using the fact
that(2.18)bi−4E(Xi′2)E(Xj′2)≤bi−2pE|Xi′|pE|Xj′|p∀j<iboth in either case (a) (2.19)∑i=2∞EXi′2bi4∑j=1i−1EXj′2≤∑i=2∞E|Xi′|pbi2p∑j=1i−1E|Xj′|p≤4∑i=2∞bi−pE(φip(|Xi|)φip(bi)+φip(|Xi|))∑j=1i−1bjpE(φjp(|Xj|)φjp(bj)+φjp(|Xj|))<∞,or case (b) (2.20)∑i=2∞EXi′2bi4∑j=1i−1EXj′2≤∑i=2∞E|Xi′|pbi2p∑j=1i−1E|Xj′|p≤4∑i=2∞bi−pE(φi(|Xi|)φi(bi)+φi(|Xi|))∑j=1i−1bjpE(φj(|Xj|)φj(bj)+φj(|Xj|))∞.Thus, we have established (2.9)
and consequently (2.7).Corollary 2.2.
Let{Xn,n≥1} be a sequence of ρ*-mixing random variables satisfying the
condition (2.21)∑i=1∞E(|Xi|rtir+|Xi|rt)<∞with r=p for 0<t<1 and all 1<p≤2,
or r=2/t for 1≤t<2.
Then,(2.22)n−1/t∑i=1n{Xi−E(XiI[|Xi|<i1/t])}⟶0a.s. asn⟶∞,Proof.
Letan=bn=n1/t for any 0<t<2.
Then, the assumption (B) of Theorem 2.1 is fulfilled.
Indeed we see(2.23)(B):∑n=1∞P[|Xn|≥an]=∑n=1∞P[|Xn|≥n1/t]≤C∑n=1∞P[2|Xn|rt≥nr+|Xn|rt]<C⁢⋅∑n=1∞E(|Xn|rtnr+|Xn|rt)<∞
by (2.21) with r=p for 0<t<1 or r=2/t for 1≤t<2.
Let now0<t<1.
Then, for φn(x)=xt, x>0, n≥1, the conditions (A) and (C) with 1<p≤2 are fulfilled: (2.24)(A):∑i=2∞bi−pE(φip(|Xi|)φip(bi)+φip(|Xi|))∑j=1i−1bjpE(φjp(|Xj|)φjp(bj)+φjp(|Xj|))≤(∑i=1∞E(|Xi|ptip+|Xi|pt))2<∞, by (2.21) with r=p. (2.25)(C):∑n=1∞E(φn2p[min⁡(|Xn|,an)]φn2p(bn)+φn2p(|Xn|))=∑n=1∞E([min⁡(|Xn|,an)]2ptn2p+|Xn|2pt)≤∑n=1∞E(|Xn|2ptn2p+|Xn|2ptI[|Xn|≥n1/t])+∑n=1∞E(|Xn|2pt|Xn|pt((np/|Xn|pt)⁢⋅np+|Xn|pt)I[|Xn|<n1/t])≤C∑n=1∞P[|Xn|≥n1/t]+∑n=1∞E(|Xn|ptnp+|Xn|pt)<∞ by (2.23) and (2.21) with r=p, 1<p≤2.
Thus, by Theorem2.1, we have(2.26)n−1/t∑i=1n{Xi−E(XiI[|Xi|<i1/t])}⟶0a.s.⁢as⁢⁢n⟶∞for 0<t<1.
Now, we need to show that (2.26) also holds for 1≤t<2.
Letφn(x)=x2.
Then, for p=2,
by the similar calculations as in case 0<t<1,
we get (2.27)(A1):∑i=2∞bi−pE(φi(|Xi|)φi(bi)+φi(|Xi|))∑j=1i−1bjpE(φj(|Xj|)φj(bj)+φj(|Xj|))≤(∑i=1∞E(|Xi|2i2/t+|Xi|2))2<∞ by (2.21) with r=2/t and (2.28)(C1):∑n=1∞E(φn2[min⁡(|Xn|,an)]φn2(bn)+φn2(|Xn|))=∑n=1∞E([min⁡(|Xn|,an)]4n4/t+|Xn|4)≤C∑n=1∞P[|Xn|≥n1/t]+∑n=1∞E(|Xn|2n2/t+|Xn|2)<∞ by (2.23) and (2.21) with r=2/t.
Therefore, by Theorem2.1, we get (2.26) for 1≤t<2.
This completes the proof of Corollary2.2.Corollary 2.3.
Let{X,Xn,n≥1} be a sequence of ρ*-mixing random variables with E|X|t<∞, 0<t<2.
Let (2.29)P[|Xi|>x]≤CP[|X|>x]for all x>0, i≥1 and some positive constant C.
Moreover, when1≤t<2, let EX=0.
Then,(2.30)n−1/t∑i=1nXi⟶0a.s. asn⟶∞.Proof.
We first note that
(2.29) implies(2.31)E(|Xi|sI[|Xi|<a])≤C{E(|X|sI[|X|<a])+asP[|X|≥a]}for any a>0 and s>0.
Put nowan=bn=n1/t for any 0<t<2.
It is easy to see thatE|X|t<∞,
(2.29) and (2.31) with s=rt imply convergence of the
series:(2.32)∑n=1∞E(|Xn|rtnr+|Xn|rt)for r=p, (1<p≤2) in the case 0<t<1 and r=2/t in the case 1≤t<2.
We have(2.33)∑n=1∞E(|Xn|rtnr+|Xn|rt)=∑n=1∞E(|Xn|rtnr+|Xn|rtI[|Xn|<n1/t])+∑n=1∞E(|Xn|rtnr+|Xn|rtI[|Xn|≥n1/t])≤∑n=1∞E(|Xn|rtnrI[|Xn|<n1/t])+∑n=1∞P[|Xn|≥n1/t]≤C{∑n=1∞E(|X|rtnrI[|X|<n1/t])+∑n=1∞P[|X|≥n1/t]}≤C{∑n=1∞E(|X|t+t(r−1)I[|X|<n1/t])n1+1/t⁢⋅t(r−1)+E|X|t}<∞.Because of Corollary 2.2, this
proves that (2.26) holds for any 0<t<2.
To complete the proof we should show
that(2.34)n−1/t|∑i=1nE(XiI[|Xi|<i1/t])|⟶0asn⟶∞for any 0<t<2.
For0<t<1 and I1:=n−1/t∑i=1ni1/tP[|X|≥i1/t], we have(2.35)n−1/t|∑i=1nE(XiI[|Xi|<i1/t])|≤n−1/t∑i=1nE(|Xi|I[|Xi|<i1/t])≤C{n−1/t∑i=1nE(|X|I[|X|<i1/t])+I1}≤C{n−1/t∑i=1nE(|X|I[|X|<n1/t])+I1}≤C{n1−1/tE(|X|I[|X|<n1/t])+I1}=C{n1−1/t∑j=1nE(|X|I[j−1≤|X|t<j])+I1}.But(2.36)∑n=1∞P[|X|≥n1/t]≤CE|X|t<∞,so by Kronecker's lemma we
get(2.37)I1:=n−1/t∑i=1ni1/tP[|X|≥i1/t]⟶0asn⟶∞.Moreover, we
note(2.38)∑j=1∞j1−1/tE(|X|I[j−1≤|X|t<j])≤∑j=1∞E(|X|I[j−1≤|X|t<j])=E|X|t<∞,and by Kronecker's
lemma(2.39)n1−1/t∑j=1nE(|X|I[j−1≤|X|t<j])⟶0asn⟶∞,which together with (2.35) and
(2.37) gives (2.34) for 0<t<1.
Let1≤t<2.
First, we will show that(2.40)lim⁡n→∞⁢P[|∑i=1nXi|>εn1/t]=0.To achieve this, we put Yi=XiI[|Xi|<n1/t] for 1≤i≤n.
Because ofEXi=0 and E|X|t<∞, we have(2.41)n−1/t|∑i=1nEYi|≤n−1/t∑i=1nE(|Xi|I[|Xi|≥n1/t])≤CE(|X|tI[|X|≥n1/t])⟶0asn⟶∞.Therefore, for large enough n,
we obtain(2.42)P[|∑i=1nXi|>εn1/t]≤∑i=1nP[|Xi|≥n1/t]+P[|∑i=1n(Yi−EYi)|>εn1/t].Hence, we need only to show
that(2.43)∑i=1nP[|Xi|≥n1/t]⟶0asn⟶∞,(2.44)P[|∑i=1n(Yi−EYi)|>εn1/t]⟶0asn⟶∞.By E|X|t<∞, we get(2.45)∑n=1∞1n∑i=1nP[|Xi|≥n1/t]≤C∑n=1∞P[|X|≥n1/t]≤CE|X|t<∞,which implies (2.43).
By Lemma1.3 and (2.23) with s=2,(2.46)∑n=1∞1nP[|∑i=1n(Yi−EYi)|>εn1/t]≤C∑n=1∞n−(1+2/t)∑i=1nEYi2≤C{∑n=1∞n−(1+2/t)∑i=1nE(X2I[|X|<n1/t])+∑n=1∞n−(1+2/t)∑i=1nn2/tP[|X|≥n1/t]}≤C{∑n=1∞n−2/t∑j=1nE(X2I[j−1≤|X|t<j])+∑n=1∞P[|X|≥n1/t]}≤C{∑k=1∞E(X2I[k−1≤|X|t<k])∑n=k∞n−2/t+E|X|t}≤C{∑k=1∞k−2/t+1E(X2I[k−1≤|X|t<k])+E|X|t}≤C{∑k=1∞kP[k−1≤|X|t<k]+E|X|t}<CE|X|t<∞which gives (2.44).
Thus, we have established that (2.40) holds true.
Equations (2.40) and (2.26) imply (2.34) which completes the proof of Corollary 2.3.Corollary2.3 gives the Marcinkiewicz SLLN
for ρ*-mixing random variables {Xn,n≥1} stochastically dominated by a random variable X for 0<t<2.
The identical result was obtained by Cai [6] as a consequence of complete
convergence theorem. Both of them, for 0<t<1,
are a special case of more general result of Fazekas and Tómács [11] obtained
for stochastically dominated random variables without assuming any kind of
dependence.



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