Asymptotics and renewal approximation in the online selection of increasing subsequence

Abstract

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, we represent the problem in terms of a controlled piecewise deterministic Markov process with decreasing paths. We apply a comparison method to the optimality equation to obtain fairly complete asymptotic expansions for the moments of the maximal length, and, with the aid of a renewal approximation, give a novel proof to the central limit theorem for the length of selected subsequence under either the optimal strategy or a strategy sufficiently close to optimality.