Asymptotic optimal probabilities for Markov-dependent last-success trials

Determine the limiting optimal probabilities for selecting the last success with a fixed number of selection opportunities in nonhomogeneous two-state Markov-dependent trials, and characterize whether analogues of the independent-trial asymptotic constants hold.

Background

The paper develops finite-horizon optimal multiple-stopping rules for nonhomogeneous two-state Markov trials under a single-crossing condition. Unlike independent Bernoulli trials, Markov dependence can produce persistent success clusters, so the independent-trial lower bounds and asymptotic constants do not automatically extend to this setting.

The paper explicitly notes that the limiting optimal probabilities for multiple selections had not been determined in the cited work and treats the possible extension of the independent-trial constants as conjectural. It subsequently proves a Poisson-limit result only under additional rare-transition and vanishing-persistence assumptions, leaving the general Markov-dependent asymptotic problem unresolved.

References

Ano, Kakie and Miyoshi proposed that analogues of the independent-trial asymptotic constants may hold for Markov-dependent trials. Their paper also notes that the limiting optimal probabilities for multiple selections were not determined. We therefore record this only as a conjectural direction from that paper and do not identify the limits with the bounds in Theorem~\ref{th32}.

— Monotone Multiple Stopping and Last-Success Problems  (2610.10237 - Ano, 7 Oct 2026) in Section 4, subsection “Last-success problem in Markov-dependent trials,” paragraph following Theorem th_markov_multiple and preceding Section “A Poisson-limit counterpart of Theorem th32”