The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case
The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case
复制标题
临界情况下的随机差分方程 $Xsb n=Asb nXsb {n-1} Bsb n$
DOI：
10.1214/aop/1024404297
复制
发表时间：
1997
影响因子：
2.3
通讯作者：
L. Elie
中科院分区：
文献类型：
--
作者：
M. Babillot;P. Bougerol;L. Elie
令 (B n , A n ) n≥1 为 i.i.d 的序列。值为 R d x R * + 的随机变量。研究了当E(log A 1 ) = 0时R d 上满足随机方程X n = A n X n - 1 + B n 的马尔可夫链。对(B 1 , A 1 )的分布不作密度假设。主要结果是马尔可夫链X n 的递推、路径的稳定性、Radon 不变量测度的存在性和唯一性以及占用次数的极限定理。结果依赖于过程的更新定理 (X n , A n ... A 1 )。
Let (B n , A n ) n≥1 be a sequence of i.i.d. random variables with values in R d x R * + . The Markov chain on R d which satisfies the random equation X n = A n X n - 1 + B n is studied when E(log A 1 ) = 0. No density assumption on the distribution of (B 1 , A 1 ) is made. The main results are recurrence of the Markov chain X n , stability properties of the paths, existence and uniqueness of a Radon invariant measure and a limit theorem for the occupation times. The results rely on a renewal theorem for the process (X n , A n ... A 1 ).