马尔可夫更新点过程中的强无记忆时间和罕见事件
Strong memoryless times and rare events in Markov renewal point processes
摘要：
设W是平稳有限状态马尔可夫更新点过程的（0，t）中的点的个数。我们推导出了W分布与复合泊松分布之间的全变差距离的界。对于任何非负随机变量ε，我们构造一个"强无记忆时间“\hat ε，使得ε-t是指数分布的，条件是{ \hat ε \leq t，ε>t}，对于每个t。这用于将马尔可夫更新点过程嵌入到另一个这样的过程中，该过程的状态空间包含表示原始过程中的内存丢失的频繁观察状态。然后，我们将W写为嵌入更新奖励过程的累积奖励，并使用Erhardsson对此量的复合泊松近似误差界。对于一个更新过程，该界以一种简单的方式依赖于更新间隔时间分布的前两个矩，以及从更新间隔时间分布相对于指数分布的Radon-Nikodym导数获得的两个常数。对于一个泊松过程，边界是0。
英文摘要：
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 ζ, we construct a ``strong memoryless time'' \hat ζsuch that ζ-t is exponentially distributed conditional on {\hat ζ\leq t, ζ>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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
