ABSTRACT

Consider a discrete-time homogeneous Markov chain with initial state i. We study the distribution of $L(j,n)$, the length of the longest consecutive visits of this chain to state j until time n. We provide two limiting theorems for $L(j,n)$ and establish asymptotics for the moment generating function of $L(j,n)$. We conclude by closing the open problem raised by the authors of [T. Konstantopoulos, Z. Liu, and X. Yang, Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials, J. Appl. Probab. 53 (2016), pp. 747–764.] by providing two large deviation principles for $L(j,n)$.

摘要

考虑具有初始状态i的离散时间齐次马尔可夫链。我们研究了分布$\mathrm{大号}(j,n)$，直到时间n为止，这条链对状态j的最长连续访问的长度。我们提供了两个极限定理$\mathrm{大号}(j,n)$并为 的矩生成函数建立渐近线$\mathrm{大号}(j,n)$. 我们以结束 [T. Konstantopoulos、Z. Liu 和 X. Yang，Laplace 变换渐近和大偏差原则在伯努利试验中的最长成功运行，J. Appl。概率。53 (2016), pp. 747–764.] 通过提供两个大偏差原则$\mathrm{大号}(j,n)$.