Polynomial convergence rates for Markov kernels under nested modulated drift conditions
Résumé
When a Markov kernel $P$ satisfies a minorization condition and nested modulated drift conditions, Jarner and Roberts provided in [JR02, Th. 3.2] an asymptotic polynomial convergence rate in weighted total variation norm of $P^n(x,\cdot)$ to the $P$-invariant probability measure $\pi$. In connection with this polynomial asymptotics, we propose explicit and simple estimates on series of such weighted total variation norms, from which an estimate for the total variation norm of $P^n(x, \cdot)-\pi$ is deduced. The proofs are selfcontained and based on the residual kernel and the Nummelin-type representation of π. No coupling technique is used.
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