On the Distribution of Time Spent by a Markov Chain at Different Levels Until Achieving a Fixed State

Abstract

In this paper we consider the question of finding a distribution of the time spent by homogeneous Markov chain $Z=(Z_k)_{k\ge 0}$ (with countable state space E) at different levels of state space until first reaching a fixed point $b\in E.$ The work consists of two parts. In the first part we show that in the general case the distribution of residence time is geometric (with weight in zero). As an example we consider a skew random walk $S^{\alpha}=(S^{\alpha}_k)_{k\ge 0}$ with parameter $\alpha\in [0,\, 1].$ In this case we obtain the distribution in explicit form. In the second part of the paper we pass to the weak limit from residence time of skew random walk to the local time of skew Brownian motion $W^{\alpha}=(W^{\alpha}_t)_{t\ge 0}$ by using the extended Donsker–Prokhorov invariance principle established in [A. S. Cherny, A. N. Shiryaev, and M. Yor, Theory Probab. Appl., 47 (2003), pp. 377–394].