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I am looking for sufficient (and optimally necessary) conditions for a discrete-time Markov chain with uncountable state space to (1) possess a unique stationary distribution with (2) exponential rate of convergence such that (3) the law of large numbers applies to any non-negative functional of finite expectation with respect to the invariant measure.

In the reference below, Poissonian resets towards an exogenous law are shown to render quite general continuous-time chains positive Harris recurrent, which suits my needs. Is it correct to deduce a state-invariant positive chance of reset at each transition of an uncountable discrete-time chain works analogously?

More generally, are discrete-time chains continuous-time ones with degenerate holding time distributions?

Avrachenkov, Konstantin; Piunovskiy, Alexey; Zhang, Yi, Markov processes with restart, J. Appl. Probab. 50, No. 4, 960-968 (2013). ZBL1295.60086.

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