关于部分和增量的Erds-Rényi大数定律的收敛速度

Convergence Rate in the Erdos-Renyi Strong Law of Large Numbers for Increments of Partial Sums

设随机变量序列列X_1,X_2,…是独立同分布的,且 EX_1=0,E exP(tX_1)<∞(t>0),S_n=X_1+X_2+…+X_n,记D_1(N,K)=max(S_(n+k)-S_n),D_2(N,K)=max max(S_(n+k)-S_n)其中 K=K_N= 0(IOgN)(N→∞),进一步若存在τ∈(0,1),使 K/LOg_τN→∞(N→∞),本文得到了当 N→∞时,对任意的δ>0,存在序列a_N使得|K_(-δ)D_1(N,K)-a_NK_((1/2)-δ)|→0 a.s.i=1,2改进了Huse等的结果.

The Erdos-Renyi law of large numbers(1970) is the first important result for asymptotic behaviors of increments of partial sums of a sequence of random variables with span [clogN]. Some generalizations have been done since then, such as convergence rate of the limit, some results when order of span g being either hi her or lower than logN. This paper aims at the generalization of these results to the case when the order of span being o (logN ) as N →∞ . In particalar, the convergence rates are studied, when the span KN is an integer sequence satisfyingKN/logrN→∞ as N→∞ for some ∈τ (0,1), The results of V.Huse and J.Steinebach (1985) are improved.