On the distribution of time spent by a Markov chain at different levels until achieving a fixed state (Q2882771)

The author considers a homogeneous Markov chain with discrete time and countable state space. The main problem is to find the distribution of the number of crossings of the level \(a\) up to the first hitting time to the state \(b\). It is named the distribution of the residence time. It is proved that the distribution of the residence time is geometric with some unknown parameters. The main result is based on the fact that the parts of Markov chain between two hittings of the same state are independent. The unknown parameter are calculated for the skew random walk. In the second part of the paper the weak limit theorem is obtained for for passage from a residence time to a local time of the skew Brownian motion. It is established that the corresponding local time of the skew Brownian motion has an exponential distribution with weight in zero.