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

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  08 March 2023

William Oçafrain
William Oçafrain*
Affiliation:
Université de Lorraine
*
*Postal address: IECL, UMR 7502, F-54000, Nancy, France. Email address: w.ocafrain@hotmail.fr
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Abstract

This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose 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 is then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin condition.

Keywords

Ergodic theoremlaw of large numberstime-inhomogeneous Markov processesquasi-stationarityquasi-ergodic distributionmoving boundaries

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60F25: $L^p$-limit theorems 60J55: Local time and additive functionals
Secondary: 60J60: Diffusion processes 60J65: Brownian motion 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 672 - 700
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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