NA序列重对数律的几个极限定理

设{X_n;n≥1}均值为零、方差有限的NA平稳序列。记S_n=∑_(k=1)~n X_k,M_n=maxk≤n|S_k|,n≥1.假设σ~2=EX_1~2+2∑_(k=2)~∞EX_1X_k>0。本文讨论了:当ε 0时,P{M_n≥εσ(2nloglogn)~(1/2)的一类加权级数的精确渐近性质,以及当ε∞时,P{M_n≤εσ(π~2n/(8loglogn))~(1/2)}的一类加权级数的精确渐近性质。这些性质与重对数律和Chung重对数律的速度有关。     

自正则Chung型重对数律的精确渐近性

设X,X_1,X_2,…为零均值、非退化、吸引域为正态吸引场的独立同分布随机变量序列,记S_n=■X_j,M_n=■|S_k|,V_n~2=■X_j~2,n≥1.证明了当b>-1时,■δ~(-2(b 1))■(log log n)~P/(n log n)P(Mn/V_n≤ε~(π~2)/(8lgo log n)~(1/2)) =4/πГ(b 1)■~(-1)~k/(2k 1)~(2b 3).     

Chung’s law of the iterated logarithm for subfractional Brownian motion

Let X~H= {X~H(t), t ∈ R_+} be a subfractional Brownian motion in R~d. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that X~H has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of X~H, we establish Chung's law of the iterated logarithm for X~H.     

Laws of the iterated logarithm for iterated Wiener processes

The recent interest in iterated Wiener processes was motivated by apparently quite unrelated studies in probability theory and mathematical statistics. Laws of the iterated logarithm (LIL) were independently obtained by Burdzy⁽²⁾ and Révész⁽¹⁷⁾. In this work, we present a functional version of LIL for a standard iterated Wiener process, in the spirit of functional asymptotic results of an ²-valued Gaussian process given by Deheuvels and Mason⁽⁹⁾ in view of Bahadur-Kiefer-type theorems. Chung's liminf sup LIL is established as well, thus providing further insight into the asymptotic behavior of iterated Wiener processes.