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

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 $\frac{1}{t} \int_0^t f(X_s)ds$ converges in $\L^2$ towards a limiting distribution, starting from any initial distribution for the process $(X_t)_{t \geq 0}$. 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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Dates and versions

hal-02963683 , version 1 (11-10-2020)
hal-02963683 , version 2 (05-04-2022)

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William Oçafrain. An ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, with application to quasi-stationarity with moving boundaries. 2022. ⟨hal-02963683v2⟩
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