Optimal Online Selection of a Monotone Subsequence: a Central Limit Theorem

Abstract

Consider a sequence of $n$ independent random variables with a common continuous distribution $F$, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy $π_n^*$ that is optimal in the sense that it maximizes the expected value of $L_n(π_n^*)$, the number of selected observations. We investigate the distribution of $L_n(π_n^*)$; in particular, we obtain a central limit theorem for $L_n(π_n^*)$ and a detailed understanding of its mean and variance for large $n$. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality $N$ where $N$ is a Poisson random variable that is independent of the sequence.