Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically

We are interested in quasi-stationarity and quasi-ergodicity when the absorbing boundary is moving. First we show that, in the moving boundary case, the quasi-stationary distribution and the quasi-limiting distribution are not well-defined when the boundary is oscillating periodically. Then we show the existence of a quasi-ergodic distribution for any discrete-time irreducible Markov chain defined on a finite space state in the fixed boundary case. Finally we use this last result to show the quasi-ergodicity in the moving boundary case.