Sharp error terms and neccesary conditions for exponential hitting times in mixing processes

• 成果类型: Article
• 署名作者: Abadi, M
• 署名单位: Universidade de Sao Paulo
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1078415835
• 发表日期: 2004
• 页码: 243-264
• 关键词: Rare events; return times; statistics; systems

摘要

We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in alpha-mixing processes with exponential decay, phi-mixing processes with a summable function phi and for general psi-mixing processes with a finite alphabet. In the first case the bound is uniform as a function of the measure of the event. In the last two cases the bound depends also on the time scale t. This allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time. A uniform bound is a consequence. We present an example that shows that this bound is sharp. We also prove that second moments are not necessary for having the exponential law. Moreover, we prove a necessary condition for having the exponential limit law.