Abstract
Consider a time-inhomogeneous Markov chain which converges in probability to a subset $S_0$ of its state space. Override the standard move mechanism up to a random transition time, almost surely finite, but not necessarily a stopping time. Under weak conditions, the modified process converges in probability to the same set $S_0$. Two examples of independent interest illustrate this result.
Citation
Bennett L. Fox. George W. Heine. "Probabilistic Search with Overrides." Ann. Appl. Probab. 5 (4) 1087 - 1094, November, 1995. https://doi.org/10.1214/aoap/1177004607
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