Limit theorems for random difference equations driven by mixing processes
Limit theorems for random difference equations driven by mixing processes
复制标题
混合过程驱动的随机差分方程的极限定理
DOI：
10.1215/kjm/1250519407
复制
发表时间：
1992
影响因子：
--
通讯作者：
Tsukasa Fujiwara
中科院分区：
文献类型：
--
作者：
Tsukasa Fujiwara
研究了由平稳混合过程驱动的随机差分方程所确定的随机过程序列的弱收敛律.关于混合过程驱动的随机过程的极限定理，包括随机常微分方程的情形，有很多关于中心极限定理和扩散逼近定理的研究。这些结果可以在Khaslminskii [15]、Keslgimov-Linnik [8]、Kesten-Papanicolaou [14]、Kesler-Kurtz [4]、Kushner [19]、Kunita [16]、[17]及其参考文献中找到。本文的工作深受这些文献的影响，但我们要强调的是，本文的一个显著特点是发展了这些文献，使极限过程具有跳跃。在这一点上，我们受到Gnedenko-Kolmogorov [7]和Samur [21]，[22]的工作的强烈激励。L e t fek; nEN，kEN * I，其中N ={1，2，···}和N *={0，1，2，···}，是定义在概率空间（Q，g，P）上的重值随机变量的n阵列。在本文中，我们假设le，？; kEN * 1对于每nEN是平稳的。L ∈ IFn（x），Gn（x）; n ∈ N I是Rd上的函数序列.对于每个ne_N，我们归纳地确定一个值随机过程{yD 7 kl; k c N *}，
T h e purpose o f this paper is to study the weak convergence of laws o f a sequence o f stochastic processes determined through random difference equations driven by stationary mixing processes. A s concerns limit theorems fo r stochastic processes driven by mixing processes, including the case of random ordinary differential equations, there a re a lot o f studies on the central limit theorem and the diffusion approximation theorem. These results can be found in Khaslminskii [15], Ibragimov-Linnik [8], KestenPapanicolaou [14], Ethier-Kurtz [4], Kushner [19], Kunita [16], [17], and many articles i n their references. T h is work is much influenced by these papers while we would like to emphasize that a notable feature o f this paper is to develop these works to allow t h e limit processes to have jumps. In this point, we a re strongly motivated by the works o f Gnedenko-Kolmogorov [7], and Samur [21], [22]. L e t fek ; nEN, kEN*I , where N = {1, 2, •••} and N *= { 0, 1, 2, •••} , be a n array of Re-valued random variables defined on a probability space (Q, g , P ) . Throughout this paper, we suppose that le,?; kEN*1 is stationary fo r every nE N . L e t IFn(x), Gn(x); n E N I be a sequence o f functions on R d . T h e n , f o r each ne_ N, we determine an valued stochastic process {yD 7 kl ; k c N *} inductively by