An ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, with application to quasi-stationarity with moving boundaries

2020/10/12 by Oçafrain, William
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2010.05483

Abstract

This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose the time-inhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function f, the time average (1)/(t) ∫₀^t f(X_s)ds converges in \mathbbL² towards a limiting distribution, starting from any initial distribution for the process (X_t)_{t ≥ 0}. This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result will be then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin's condition.

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