Strong memoryless times and rare events in Markov renewal point processes

• 成果类型: Article
• 署名作者: Erhardsson, T
• 署名单位: Royal Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000054
• 发表日期: 2004
• 页码: 2446-2462
• 关键词: compound poisson approximation; chains

摘要

Let W be the number of points in (0, t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable (, we construct a strong memoryless time zeta such that zeta - t is exponentially distributed conditional on {zeta less than or equal to t, zeta > t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon-Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.