In this paper,a liminf behavior is studied of a two-parameter Gaussian process which is a gener-alization of a two-parameter Wiener process.The results improve on the liminfs in [7].
The strong approximations of a class of R^d-valued martingales are considered. The conditions used in this paper are easier to check than those used in [3] and [9]. As an application, the strong approximation of a class of non-homogenous Markov chains is established, and the asymptotic properties are established for the multi-treatment Markov chain adaptive designs in clinical trials.
设{X,Xn;n≥1)为i.i.d.的随机变量序列,其均值为0且EX2=1.令S={Sn}n≥0为一维随机游动,其中S0=0,Sn=sum from k=1 to n Xk,对n≥1.定义G(n)为随机游动局部时的Cauchy主值.本文得到了,若存在某δ1>0,E|X|2r/(3p-4)+δ1<∞成立,那么对4/3P。
Let {X, X; ∈Nd} be a field of i.i.d, random variables indexed by d-tuples of positiveintegers and taking values in a Banach space B and let X<sup>（r）=Xm if ‖X‖ is the r-th maximum of{‖X‖; ≤. Let S=∑（≤）Xand （r）S=S-(X<sup>（1）+…+X<sup>（r）. We approximate the trimmedsums （r）n, by a Brownian sheet and obtain sufficient and necessary conditions for （r）Sto satisfy thecompact and functional laws of the iterated logarithm. These results improve the previous works byMorrow （1981）, Li and Wu （1989） and Ledoux and Talagrand （1990）.
Li Xin ZHANG Department of Mathematics Xixi Campus. Zhejiang University, Hangzhou 310028, P. R. China
