Tweedie-type stability estimates for the invariant probability measures of perturbed Markov chains under drift conditions
Résumé
Given a perturbed version $P_{\theta}$ of a Markov kernel $P_{\theta_0}$ with respective invariant probability measures $\pi_{\theta}$ and $\pi_{\theta_0}$ , we provide estimates of the $W$-weighted norm $\|\pi_\theta - \pi_{\theta_0}\|_{W}$ for some Lyapunov function $W$. We follow Tweedie's approach proposed in a seminal paper on the truncation-augmentation scheme for approximating the invariant probability measure of discrete-state Markov kernels. But the novelty here is that the state space for the Markov kernels and the form of the perturbation are general, and that the intermediate term $\|\pi_\theta - P_\theta^{\, n}(x,\cdot)\|_W$ usually involved to control the error norm $\|\pi_\theta - \pi\|_{W}$ is replaced with $\|\pi_\theta - \widetilde{\mu}_n^{(\theta)}\|_{W}$, where $\widetilde{\mu}_n^{(\theta)}$ is an alternative probability measure which has been introduced in a recent work for approximating $\pi_{\theta_0}$ under a minorization condition.The interest is that the estimates of $\|\pi_\theta - \widetilde{\mu}_n^{(\theta)}\|_{W}$ turn out to be much more accurate and practicable than for $\|\pi_\theta - P_\theta^{\, n}(x,\cdot)\|_W$ under geometric or polynomial drift conditions. Moreover we do not need to resort to the use of techniques related to the existence of an atom for $P_{\theta_0}$. This study is performed for geometrically or polynomially ergodic Markov kernels, and compared with prior works when applied to the standard truncation-augmentation scheme for discrete-state Markov kernels.
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