NA随机变量对数律的收敛速度

Convergence Rate in the Law of Logarithm for NA Random Variables

设{X,X_(n),n>1}是同分布的NA随机变量序列,h(·)>0是定义在(0,∞)上的不减函数且满足∫_(1)^(∞)[th(t)^(-1)dt=∞.令ψ(t)=∫_(1)^(t)[sh(s)]^(-1)ds,t≥1,S_(n)=Σ_(i=1)^(n)X_(i),n≥1,Lt=ln max{e,t}.本文证明了Σ_(n)^(∞)=1[nh(n)]^(-1)P(max_(1)≤j≤n|S_(j)|≥(1+ε)σ√2nLψ(n))<∞,∀ε>0成立的充要条件是E(X)=0和E(X^(2))=σ^(2)∈(0,∞).这一结果部分地推广了文献[7]的结论.

Let{X,X_(n),n>1}be a sequence of identically distributed NA random variables and set S(n)=Σ_(i=1)^(n)X_(i),n≥1.Let h(·)be a positive nondecreasing function on(0,∞)such that∫_(1)^(∞)[th(t)^(-1)dt=t∞.Denote Lt=lnmax{e,t},S(n)=Σ_(i=1)^(n)X_(i),ψ(t)=∫_(1)^(t)[sh(s)]^(-1)ds,t≥1.In this paper,we prove thatΣ_(n)^(∞)=1[nh(n)]^(-1)P(max_(1)≤j≤n|S_(j)|≥(1+ε)σ√2nLψ(n))<∞,∀ε>0 if and if E(X)=0 and E(X^(2))=σ^(2)∈(0,∞).The result partially extends and improves the theorems of[7].