NA RANDOM VARIABLES

NA RANDOM VARIABLES

GAN SHIXIN

Received 5 July 2003 and in revised form 1 April 2004

We study the almost sure (strong) stability of weighted sums of NA random variables and obtain some new results which extend earlier results of Matula (1992), Chow and Teicher (1971), Jamison et al. (1965), and Petrov (1975).

1. Introduction and preliminaries

We start with definitions. Let (Ω,Ᏺ,P) be a probability space. The random variables we deal with are all defined on (Ω,Ᏺ,P). A random variable sequence{Y_n,n≥1}is said to be strongly stable if there exist two constant sequences{bn}and{dn}with 0< bn↑ ∞ such that

b⁻_n¹Y_n−d_n−→0 a.s. (1.1)

A random variable sequence{Xn,n≥1}is said to be stochastically dominated by a nonnegative random variableX(write{X_n}< X) if there exists a constantc >0 such that PXn> t≤cP(X > t) ∀t >0,∀n≥1. (1.2) A finite family of random variables{X_i, 1≤i≤n}is said to be negatively associated (abbreviated to NA) if for any disjoint subsetsAandBof{1, 2,. . .,n}and any real coor- dinatewise nondecreasing functions f onR^AandgonR^B,

CovfX_i,i∈A,gX_j,j∈B≤0 (1.3) whenever the covariance exists. An infinite family of random variables{Xi,i≥1}is NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan [5].

They also pointed out and proved in their paper that a number of well-known multivari- ate distributions possess the NA property. Now people know that NA random variables have wide application in reliability theory and multivariate statistical analysis. Recently Su et al. [12] show that NA structure plays an important role in risk management. Because of these reasons, the notions of NA random variables have received more and more atten- tion in recent years. A great number of papers for NA random variables have appeared in

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:6 (2005) 975–985 DOI:10.1155/IJMMS.2005.975

the literature. We refer to Joag-Dev and Proschab [5] for fundamental properties, New- mann [8] for the central limit theorem, Matula [7] for the three-series theorem, Shao and Su [11] for the law of the iterated logarithm, Shao [10] for moment inequalities, Liu et al.

[6] for the H`ajek-R`enyi inequality, and Barbour et al. [1] for Poison approximation.

The main purpose of this paper is to study the strong stability of weighted sums of NA random variables and try to obtain some new results which extend the corresponding re- sults of Matula [7], Chow and Teicher [2], Jamison et al. [4], and Petrov [9]. For this goal, we need some lemmas. The following lemma is a simple extension of [7, Theorem 3].

Lemma1.1. Let{X_n,n≥1}be a sequence of NA random variables with finite second mo- ments and{bn,n≥1}a sequence of real numbers with0< bn↑ ∞. If^∞_n=1Var(Xn)/b²_n<∞, then^∞_n=1(Xn−EXn)/bnconverges a.s., and thereforeⁿ_i=1(Xi−EXi)/bn→0a.s.

Note that{Xn/bn,n≥1}is a sequence of NA random variables by [5, property P6]. By using [7, Theorem 3], we can immediately obtainLemma 1.1.

For a random variableX, writeX^(c)=XI(|X| ≤c) +cI(X > c)−cI(X <−c) for some c >0.

Lemma1.2 (see [7, Theorem 4]). Let{Xn,n≥1}be a sequence of NA random variables. If, for somec >0, the series^∞_n=1EXn^(c),^∞_n=1Var(Xn^(c)), and^∞_n=1P(|X_n| ≥c)are convergent, then^∞_n=1Xnis convergent a.s.

Lemma1.3. LetXbe a random variable andX0a nonnegative random variable. If, for any t >0,P(|X|> t)≤cP(X0> t), then for allp >0,t >0,

E|X|^pI|X| ≤t≤ct^pPX0> t+EX₀^pIX0≤t. (1.4) Proof. By the integral equality

p _t

0s^p−1P|X|> sds=t^pP|X|> t+E|X|^pI|X| ≤t, (1.5) it follows that

E|X|^pI|X| ≤t≤p _t

0s^p−1P|X|> sds≤cp _t

0s^p−1PX0> sds

=ct^pPX0> t+EX₀^pIX0≤t.

(1.6) Lemma1.4. Let{an,n≥1}and{bn,n≥1}be two sequences of positive numbers withcn= b_n/a_n andb_n↑ ∞. Let {X_n,n≥1}be a sequence of mean zero variables with{X_n}< X, whereXis a nonnegative random variable. DefineN(x)=Card{n:cn≤x},x >0. If

(1)^∞_i=1P(|Xi|> ci)<∞,

(2)^∞_n=11^∞P(|X_n|> sc_n)ds <∞, or (1)EN(X)<∞,

(2)₁^∞EN(X/s)<∞, then b⁻_n¹

n i=1

aiEX_i^(ci)−→0 asn−→ ∞. (1.7)

Proof. Obviously (1)⇒(1) and (2)⇒(2). It suffices to show that under conditions (1) and (2) we have (1.7):

∞ i=1

aiEX_i^(ci)

bi =

∞ i=1

c⁻_i¹EXi−ci

IXi> ci

+EXi+ci

IXi<−ci

≤ ∞ i=1

c⁻_i¹EX_i+c_iIX_i> c_i

= ∞ i=1

c⁻_i¹EXiIXi> ci +

∞ i=1

PXi> ci ,

(1.8)

∞ i=1

c⁻_i¹EXiIXi> ci

≤ ∞ i=1

c⁻_i¹

ciPXi> ci

+ _∞

ci

PXi> tdt

=^∞

i=1

PX_i> c_i+ ∞ i=1

_∞

1 PX_i> sc_ids <∞.

(1.9)

Therefore, from (1.8) and (1.9),^∞_i=1aiEX_i^(ci)/bi converges. By Kronecker’s lemma, we

have (1.7).

Corollary1.5. Let{X_n,n≥1}be a sequence of mean zero random variables with{X_n}<

X∈Llog⁺L. If

(1)^∞_n=1P(|Xn|> n)<∞, (2)^∞_n=1

_∞

1 P(|X_n|> sn)ds <∞, then n⁻¹

n i=1

EX_i⁽ⁱ⁾−→0 asn−→ ∞. (1.10)

Proof. Clearlycn=n,n≥1. The first part of proof is like inLemma 1.4. We only give the last part of proof:

∞ i=1

i⁻¹EXiIXi> i

≤ ∞ i=1

i⁻¹

iPXi> i+ _∞

i PXi> tdt

(1.11)

≤c ∞ i=1

P(X > i) +c ∞ i=1

i⁻¹ _∞

i P(X > t)dt

≤cEX+c ∞ i=1

i⁻¹ ∞ k=i

P(X > k)≤cEX+c ∞ k=1

k i=1

i⁻¹P(X > k)

≤cEX+c ∞ k=1

(1 + logk)P(X > k)≤cEX+cEXlog⁺X <∞.

(1.12)

This shows thatn⁻¹ⁿ_i=1EX_i⁽ⁱ⁾→0 asn→ ∞.

Throughout this paper, the symbolcstands for a generic positive constant which may differ from one place to another.

2. Strong stability

Theorem2.1. Let{Xn,n≥1}be a sequence of NA random variables and{gn(x),n≥1} a sequence of even functions, positive and nondecreasing in the intervalx >0. If one of the following conditions is satisfied for everyn≥1:

(1)x/gn(x)↑as0< x↑,

(2)x/g_n(x)↓andg_n(x)/x²↑as0< x↑and alsoEX_n=0,

then for any positive real-number sequence{bn,n≥1}withbn↑ ∞satisfying ∞

n=1

Egn Xn gn

bn <∞, (2.1)

the series^∞_n=1Xn/bnconverges almost surely, and thereforeⁿ_i=1Xi/bn→0a.s.

Proof. For eachn≥1, putZn=Xn/bn. Then{Zn,n≥1}remains a sequence of NA ran- dom variables by [5, property P6]. Takec=1 inLemma 1.2. Sinceg_n(x)↑asx >0, then gn(|Xn|)≥gn(bn) on{|Zn| ≥1}. So

∞ n=1

PZn≥1≤ ∞ n=1

g_nX_nIZ_n≥1 gn

bn

dP≤ ∞ n=1

Eg_nX_n gn

bn

<∞. (2.2)

We will suppose that the functiongn(x) satisfies condition (1), then, in the interval|x| ≤ bn, we havex²/b²_n≤g_n²(x)/g_n²(bn)≤gn(x)/gn(bn). If, however,nis such that (2) is satisfied, then, in the same interval, we havex²/b_n²≤g_n(x)/g_n(b_n), therefore

∞ n=1

EZ_n⁽¹⁾²= ∞ n=1

Z²_nIZn≤1dP+

IZn>1dP

≤ ∞ n=1

gn

Xn

IXn≤bn

gn

bn dP+

gn

Xn

IXn> bn

gn

bn dP

= ∞ n=1

Egn

Xn

gn

bn <∞.

(2.3)

If condition (1) is satisfied, then EZ_n⁽¹⁾≤ XnIXn≤bn

bn dP+

IXn> bn

dP

≤

g_nX_nIX_n≤b_n g_nb_n dP+

g_nX_nIX_n> b_n g_nb_n dP

=Egn Xn g_nb_n .

(2.4)

If condition (2) is satisfied, then EZ_n⁽¹⁾=

Zn−1IZn>1dP+ Zn+ 1IZn<−1dP

≤ Z_nIZ_n>1dP+

IZ_n>1dP

≤

gnXnIZn>1 gn

bn dP+

gnXnIZn>1 gn

bn dP

≤2Eg_nX_n gn

bn

.

(2.5)

Consequently, we have ∞ n=1

EZ_n⁽¹⁾≤2 ∞ n=1

Egn

Xn

gn

bn

<∞. (2.6)

Thus (2.2), (2.3), (2.6), andLemma 1.2imply the convergence of^∞_n=1Z_n=_∞

n=1X_n/b_n almost surely.

Takegn(x)= |x|^p,n≥1, 1≤p≤2, inTheorem 2.1. We can infer the following impor- tant special case which is a further extension ofLemma 1.1.

Corollary 2.2. Let {X_n,n≥1} be a sequence of NA mean zero random variables and {bn,n≥1}a sequence of real numbers with0< bn↑ ∞,1≤p≤2. If^∞_n=1E|Xn|^p/bn^p<∞, thenⁿ_i=1X_i/b_n→0a.s.

Theorem2.3. Let{a_n,n≥1}and{b_n,n≥1}be two sequences of positive numbers with cn=bn/an andbn↑ ∞. Let{Xn,n≥1}be a sequence of NA random variables which is stochastically dominated by a nonnegative random variableX. SetN(x)=Card{n:cn≤x}, x >0.1≤p≤2. If the following conditions are satisfied:

(1)EN(X)<∞,

(2)₀^∞t^p−1P(X > t)_t^∞N(y)/ y^p+1d ydt <∞, then there existd_n∈R,n=1, 2,. . .such that

b⁻_n¹ n i=1

aiXi−dn−→0 a.s. (2.7)

Proof. LetSn=_n

i=1aiXi,Tn=_n

i=1aiX_i^(ci),n≥1. Obviously we have ∞

i=1

PXi=X_i^(ci)= ∞ i=1

P|Xi|> ci

≤c ∞ i=1

PX > ci

≤cEN(X)<∞. (2.8)

By Borel-Cantelli lemma for any sequence{dn} ⊂R, the sequences{b⁻_n¹Tn−dn} and {b⁻_n¹S_n−d_n}converge on the same set and to the same limit. We will show thatb⁻_n¹ⁿ_i=1a_i (X_i^(ci)−EX_i^(ci))→0 a.s. which gives the theorem withdn=b⁻_n¹ⁿ_i=1aiEX_i^(ci). Now note that {a_i(X_i^(ci)−EX_i^(ci)),i≥1} is a sequence of NA mean zero random variables by

[5, property P6]. It follows fromc_r-inequality andLemma 1.3that

∞ n=1

Ean

Xn^(cn)−EXn^(cn)p bn^p

≤c ∞ n=1

c⁻n^pEXn^pIXn≤cn

+c ∞ n=1

PXn> cn

≤c ∞ n=1

c⁻n^p

cn^pPX > c_n+EX^pIX≤c_n+c ∞ n=1

PX > c_n

≤cEN(X) +c ∞ n=1

pc⁻n^p

_cn

0 t^p−1P(X > t)dt,

(2.9)

∞ n=1

pc⁻n^p

_cn

0 t^p−1P(X > t)dt=p _∞

0 t^p−1P(X > t)

{n:cn>t}

c⁻n^pdt

≤p² _∞

0 t^p−1P(X > t) _∞

t

N(y) y^p+1 d y dt.

(2.10)

The last inequality 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^−pd N(y)

=lim

u→∞

u^−pN(u)−t^−pN(t) +

t<y≤uy^−(p+1)N(y)d y

(2.11)

and u^−pN(u)≤p_u^∞y^−(p+1)N(y)d y→0 asu→ ∞. Hence, ^∞_n=1|an(Xn^(cn)−EXn^(cn))|^p/ bn^p<∞. ByCorollary 2.2, it follows thatb⁻_n¹ⁿ_i=1a_i(X_i^(ci)−EX_i^(ci))→0 a.s. This completes

the proof ofTheorem 2.3.

Remark 2.4. Heyde’s [3, Theorem 2] extended [4, Theorem 2] of Jamsion et al. to more general weights, which are just the same weights considered in our paper. SoTheorem 2.3 is an extension of [3, Theorem 2].

FromTheorem 2.3andLemma 1.4, we have the following corollaries.

Corollary 2.5. Let the conditions of Theorem 2.3be fulfilled and EX_n=0,n≥1, and _∞

1 EN(X/s)ds <∞. Thenb_n⁻¹ⁿ_i=1aiXi→0a.s.

Corollary 2.6. Let {Xn,n≥1} be a sequence of NA mean zero random variables and {Xn}< X.

(1)IfX∈Llog⁺L, thenn⁻¹ⁿ_i=1X_i→0a.s.

(2)IfX∈L^r, 1< r <2, thenn^−1/rn_i=1X_i→0a.s.

Proof. (1) Takea_n=1,b_n=n,n≥1,p >1, inTheorem 2.3, thenc_n=n,N(x)=Card{n: c_n≤x} ≤x,x≥0. It is easy to show that conditions (1) and (2) inTheorem 2.3hold true.

Theorem 2.3andCorollary 1.5guarantee thatn⁻¹ⁿ_i=1Xi→0 a.s.

(2) Takean=1,bn=n^1/r,n≥1,p=2, inTheorem 2.3. Putδ=2−r >0, thenN(x)= Card{n:n^1/r≤x} ≤x^r,x≥0. It is easy to verify that the conditions inCorollary 2.5hold true. ByCorollary 2.5, we haven^−1/rn_i=1Xi→0 a.s.

Theorem2.7. Let{an,n≥1}and{bn,n≥1}be two sequences of positive numbers with cn=bn/an andbn↑ ∞. Let{Xn,n≥1}be a sequence of NA random variables which is stochastically dominated by a nonnegative random variableX.N(x)=Card{n:c_n≤x}, x >0.1≤p≤2. If the following conditions are satisfied:

(1)EN(X)<∞, (2)₁^∞EN(X/s)ds <∞,

(3) max1_≤j≤nc^p_j^∞_j=nc⁻_j^p=O(n), then

b⁻_n¹ n i=1

a_iX_i−→0 a.s. (2.12)

Proof.

∞ n=1

PXn=X_n^(cn)= ∞ n=1

PXn> cn

≤c ∞ n=1

PX > cn

≤cEN(X)<∞. (2.13)

By Borel-Cantelli lemma, it suffices to show thatb⁻_n¹ⁿ_i=1a_iX_i^(ci)→0 a.s. FromLemma 1.4, we need only to show thatb⁻_n¹ⁿ_i=1ai(X_i^(ci)−EX_i^(ci))→0 a.s. Note that {ai(X_i^(ci)− EX_i^(ci)),i≥1}is a sequence of NA mean zero random variables.

∞ n=1

Ean

Xn^(cn)−EXn^(cn)p bn^p

≤c ∞ n=1

c⁻n^pEX_n^(cn)p=c ∞ n=1

c⁻n^p

EX_n^pIX_n≤c_n+cn^pPX_n> c_n

≤c ∞ n=1

c⁻n^p

cn^pPX > cn

+EX^pIX≤cn +c

∞ n=1

PX > cn

≤c ∞ n=1

PX > cn

+c ∞ n=1

EX^pIX≤cn

cn^p

.

(2.14)

Clearly

∞ n=1

PX > cn

≤EN(X)<∞. (2.15)

Putε_n=max1≤j≤nc_j,ε0=0, then ∞

n=1

EX^pIX≤cn cn^p ≤

∞ n=1

EX^pIX≤εn

cn^p =

∞ n=1

n j=1

EX^pIεj−1< X≤εj cn^p

≤ ∞ j=1

Pεj−1< X≤εj ε^p_j

∞ n=j

1 cn^p ≤c

∞ j=1

jPεj−1< X≤εj

=c ∞ j=1

j n=1

Pεj−1< X≤εj

=c ∞ n=1

PX > εn−1

≤c

1 + ∞ n=1

PX > cn <∞.

(2.16)

ByCorollary 2.2, we haveb⁻_n¹ⁿ_i=1ai(X_i^(ci)−EX_i^(ci))→0 a.s. The proof is complete.

Afterwards letα(x) :R+→R+ be a positive, nonincreasing function withan=α(n), b_n=_n

i=1a_i,c_n=b_n/a_n,n≥1, where

b_n−→ ∞, (2.17)

0<lim inf

n→∞ n⁻¹c_nαlogc_n≤lim sup

n→∞ n⁻¹c_nαlogc_n<∞, (2.18) xαlog⁺xis nondecreasing forx >0. (2.19) Theorem2.8. Let{X_n,n≥1}be a sequence of identically distributed NA random variables.

IfE|X1|α(log⁺|X1|)<∞, then there exist dn∈R,n=1, 2,. . ., such thatb⁻_n¹ⁿ_i=1aiXi− dn→0a.s.

Proof. Since 0< α(x)↓,bn↑ ∞, thencn↑ ∞. By (2.18) we can choosem0∈N,γ >0,β >0, such that, forn≥m0,

γn≤c_nαlogc_n≤βn. (2.20)

Hence, forn≥m≥m0, we havecn≥γn(α(logcm))⁻¹which guarantees that ∞

j=m

c⁻_j²≤α²logcm

γ²m , m≥m0. (2.21)

Note that {X_i^(ci),i≥1} is a sequence of NA random variables by [5, property P6]. For m≥m0,

∞ j=m

EX^(c_j ^j)2 c²_j ⁼

∞ j=m

c⁻_j²EX²_jIX_j≤c_j+ ∞ j=m

PX_j> c_j. (2.22)