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

TL;DR

This paper establishes an ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, demonstrating convergence of time averages and applying results to quasi-stationarity with moving absorbing boundaries.

Contribution

It introduces an ergodic theorem for asymptotically periodic Markov processes and applies it to quasi-stationary distributions with moving boundaries.

Findings

Time averages converge in L^2 for the processes studied.

Almost sure convergence of time averages under additional conditions.

Existence of quasi-ergodic distribution for processes 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 $\mathbb{L}^2$ 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.