COMPLETE CONVERGENCE FOR NEGATIVELY DEPENDENT RANDOM VARIABLES

COMPLETE CONVERGENCE FOR NEGATIVELY DEPENDENT RANDOM VARIABLES

M. AMINI D.

Sistan and Baluchestan University, Department of Mathematics Faculty of Science, Zahedan, Iran

E-mail: [email protected] and

A. BOZORGNIA

Ferdowski University, Department of Statistics Faculty of Mathematical Science, Mashhad, Iran

E-mail: [email protected]

(Received February 2001; Revised January 2003)

In this paper, we study the complete convergence for the means ¹

n

Pn

i=1X_i and

1 n^α

Pn

k=1X_nk via. exponential bounds, whereα >0 and{X_n, n≥1}is a sequence of negatively dependent random variables and{X_nk, 1≤k≤n, n≥1}is an array of rowwise pairwise negatively dependent random variables.

Key words: Complete Convergence, Negatively Dependent.

AMS (MOS) subject classification: 60E15, 60F15.

1 Introduction

Let {Xn, n ≥1} be a sequence of i.i.d., real random variables. Hsu and Rabbins [5]

proved that ifE(X) = 0 andE(X²)<∞, then the sequence _n¹Pn

i=1Xi converges to 0 completely. (i.e., the seriesP^∞

n=1P[|Sn|> nε]<∞, converges for everyε >0). Now let {Xn, n≥1} be a sequence of negatively dependent real random variables. In this pa- per, we proved the complete convergence of the sequence ¹_nPn

i=1Xi, via. exponential bounds. In addition if {Xnk, 1 ≤ k≤ n, n≥1} is an array of rowwise pairwise neg- atively dependent random variables, we proved complete convergence of the sequence { ¹

n^α

Pn

k=1X_nk, n≥1} whereα >0. To prove these theorems we need to the following definitions and lemmas.

Definition 1:The random variables X1,· · ·, Xn are pairwise negatively dependent if

P(Xi≤xi, Xj ≤xj)≤P(Xi≤xi)P(Xj≤xj), (1.1) 121

for all xi, xj ∈IR,i6=j. It can be shown that (1.1) is equivalent to

P(X_i> x_i, X_j > x_j)≤P(X_i> x_i)P(X_j> x_j), (1.2) for all xj, xi∈IR , i6=j.

Definition 2:The random variables X1,· · ·, Xn are said to be negatively depen- dent (ND) if we have

P(∩ⁿ_j=1(Xj≤xj))≤ Yn

j=1

P(Xj ≤xj), (1.3)

and

P(∩ⁿ_j=1(Xj> xj))≤ Yn

j=1

P(Xj > xj), (1.4) for all x₁,· · ·, x_n ∈IR. An infinite sequence {X_n, n ≥1} is said to be ND if every finite subset {X₁,· · ·, X_n} is ND.

Conditions (1.3) and (1.4) are equivalent for n= 2. However Ebrahimi and Ghosh [4] show that these definitions do not agree forn≥3.

Definition 3: The sequence {X_n, n ≥ 1} of random variables converges to zero completely (denoted limn→∞Xn= 0 completely), if for everyε >0

X∞

n=1

P[|X_n|> ε]<∞. (1.5)

Lemma 1: (Petrov [8])Let X be a random variable with E(X) = 0, E(X²)<∞, and suppose there exists a positive constant H such that for all m≥2

|E(X^m)| ≤1

2m!σ²H^m−2, (1.6)

then for every |t| ≤_2H¹

Ee^tX ≤e^t2σ2.

Lemma 2: (Serfeling [9]) Let X be a r.v. with E(X) = µ. If P[a ≤X ≤ b] = 1. Then for every real number h >0,

Ee^h(X−µ)≤e^{h2 (b−a)28} , and

Ee^h|X−µ|≤2e^{h2 (b−a)28} .

The next three lemmas will be needed in the proofs of the strong law of large numbers in the next section [3].

Lemma 3: Let X1,· · ·, Xn be ND random variables and f1,· · ·, fn be a sequence of Borel functions which all are monotone increasing (or all are monotone decreasing), then f1(X1),· · ·, fn(Xn) are ND random variables.

Lemma 4: Let X1,· · ·, Xn be pairwise ND random variables, then E(XiXj)≤E(Xi)E(Xj), ∀ i6=j.

Lemma 5: Let X1,· · ·, Xn be ND nonnegative random variables, then

E[

Yn

j=1

Xj]≤ Yn

j=1

E[Xj].

2 Exponential Bounds and Complete Convergence

In this section, we obtained some exponential bounds for probability P[|Sn| > x] for every x > 0 using Lemmas 1 and 2, and then we proved the complete convergence of the sequence {_n¹Pn

i=1Xi}. We shall consider a sequence of ND random variables {Xn, n≥1}, with zero means and finite variances . We put

Sn = Xn

k=1

Xk, Bn = Xn

k=1

σ_k².

Theorem 1: Let {Xn, n≥1} be a sequence of ND r.v.’s and suppose there exists a positive constantH such that for all m≥2and 1≤k≤n,

|E(X_k^m)| ≤ 1

2m!σ_k²H^m−2, (2.7)

ifP∞

n=1exp[−ⁿ_4B^2ε2

n]<∞, for everyε >0, then 1

n Xn

k=1

Xk−→0, completely.

Proof: By Lemmas 1, 3, 5 and Markov’s inequality for every |t| ≤ _2H¹ we have P[|S_n| ≥x]≤P[S_n≥x] +P[−S_n≥x]≤e^−txEe^tSn+e^−txEe^−tSn

≤e^−tx( Yn

k=1

Ee^tXk+ Yn

k=1

Ee^−tXk)≤2 exp[−tx+t²Bn].

Hence

P[|S_n| ≥x]≤2 exp[−tx+t²B_n]. (2.8) With h(t) =t²Bn−tx and 0≤x≤ ^B_Hⁿ, the equation h⁰(t) = 0 has the unique solution t= _2B^x

n which minimize h(t). Hence P[|Sn| ≥x]≤2 exp[− x²

4B_n] if 0≤x≤ Bn

H .

Leta=^B_H^n?, wheren^?is the first subscript so thatBn >0. Then for every 0< ε≤a, and by the assumption

X∞

n=1

P[|Sn| n ≥ε]≤

X∞

n=1

2 exp[−n²ε² 4Bn

]<∞,

and for eachε⁰> a≥ε >0, we have [

X∞

n=1

P[|Sn|

n ≥ε⁰]≤ X∞

n=1

P[|Sn|

n ≥ε]<∞.

These complete the proof.

Remark 1: In particular ifB_n=O(n^α), 0< α <2, then seriesP^∞

n=1exp[−ⁿ_4B^2ε2

n] converges.

Remark 2: If the random variables X1, X2,· · ·, Xn are ND r.v.’s with zero means and uniformly bounded, that is if there exists a positive constant c such that

P[|X_k| ≤c] = 1, k≥1 then for all integers m≥2 we have

|E(X_k^m)| ≤c^m−2σ²_k.

Thus Condition (1.6) in Lemma 1 is satisfied with H =c. Hence ifP^∞

n=1exp[−ⁿ_4B^2ε2

n]<

∞, for everyε >0, then 1 n

Xn

k=1

Xk−→0, completely.

Theorem 2: Let {Xn, n≥1}be a sequence of ND random variables and c_n = max{esssup^√|Xk|

Bn,1≤k≤n}. If P^∞

n=1exp[−_c^2nε₂ ²

nB_n]<∞, then for each ε >0, 1

n Xn

k=1

Xk−→0, completely.

Proof: By Lemmas 2, 3, 5 and Markov’s inequality for every t >0, we have P[|Sn|> ε]≤P[Sn> ε] +P[−Sn> ε]≤e^−√tεBnEe^√tSnBn +e^−√tεBnEe

−√tSn Bn

≤e^−√tεBn{ Yn

k=1

(Ee^√tXkBn +Ee

−√tXk

Bn )} ≤2 exp[− tε

√ Bn

+nt²c²_n 2 ].

Thus, for t= _nc2^ε

n

√

B_n we have

P[|Sn|> ε]≤2 exp[− ε² 2nc²_nBn

], and by the assumption we have

X∞

n=1

P[|Sn|

n > ε]≤2 X∞

n=1

exp[− nε² 2c²_nBn

]<∞ which completes the proof.

Remark 3: In particular ifc²_nBn =O(n^α), 0< α <1, then seriesP^∞

n=1exp[−_2c^nε2

nB_n] converges.

3 Strong Limit Theorem for arrays

Let {X_nk, 1 ≤k ≤n, n ≥1} be an array of rowwise pairwise ND random variables with

E[X_nk] = 0, σ²_nk=E[X_nk² ], 1≤k≤n, n≥1.

We consider the means ξn = _n¹α

Pn

k=1Xnk, n ≥ 1 where α is a fixed positive real number. SinceX_nk, 1≤k≤nare pairwise ND random variables, by Lemma 4 we can write

E[ξ_n²]≤ 1 n^2α

Xn

k=1

σ²_nk, (3.9)

because

E[ξ_n²] =E[ 1 n^α

Xn

k=1

X_nk]²= 1 n^2α

Xn

k=1

Xn

j=1

E[X_nkX_nj]

= 1 n^2α[

Xn

k=1

E[X_nk² ] +X X

k6=j

E[XnkXnj]]≤ 1 n^2α

Xn

k=1

σ_nk² .

Theorem 3: Let {Xnk, 1 ≤k≤n, n≥1} be an array of rowwise pairwise ND random variables withE[Xnk] = 0. If for some α >0

X∞

n=1

1 n^2α

Xn

k=1

σ_nk² <∞,

then

1 n^α

Xn

k=1

Xnk−→0 completely.

Proof: By Chebyshev’s inequality and (3.9), we have P[|ξn|> ε]≤ 1

ε²E[ξ_n²]≤ 1 ε²n^2α

Xn

k=1

σ²_nk.

Since for some α >0 X∞

n=1

P[|ξn|> ε]≤ X∞

n=1

1 ε²n^2α

Xn

k=1

σ²_nk<∞,

by Definition 3

1 n^α

Xn

k=1

Xnk−→0 completely.

Corollary 1: Under assumptions of Theorem 3, let σnk ≤ σkk, k ≥ 1, n ≥ (k+ 1). If P^∞

k=1 σ²_kk

k^2α−1 <∞ for some α > ¹₂ then 1

n^α Xn

k=1

Xnk−→0 completely.

Proof: We have X∞

n=1

E[ξ²_n]≤ X∞

n=1

1 n^2α

Xn

k=1

σ_nk² ≤ X∞

n=1

1 n^2α

Xn

k=1

σ_kk²

= X∞

k=1

σ²_kk X∞

n=k

1

n^2α =O(1) X∞

k=1

σ_kk² k^2α−1,

then ∞

X

n=1

P[|ξn|> ε]≤O(1) X∞

k=1

σ_kk² k^2α−1 <∞, which completes the proof.

Remark 4: The weaker conditionP^∞

k=1 σ²_kk

k^2α <∞, for everyα >0, implies only the complete convergence of subsequence {ξ₂p, p= 0,1,2, ....}, since

X∞

p=0

E[ξ₂²p]≤ X∞

p=0

1 2^2αp

2^p

X

k=1

σ²_kk

= X∞

k=1

σ²_kk X

p:2^p≥k

1

2^2αp =O(1) X∞

k=1

σ²_kk k^2α.

Acknowledgments

The authors would like to thank the referee for his careful reading of the manuscript and for many valuable suggestions which improved the presentation of the paper.

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