The random difference equation Xn = AnXn-1+Bn in the critical case

• 成果类型: Article
• 署名作者: Babillot, M; Bougerol, P; Elie, L
• 署名单位: Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 478-493
• 关键词: limit

摘要

Let (B-n, A(n))(n greater than or equal to 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)).