Optimal online selection of a monotone subsequence: a central limit theorem

Consider a sequence of nn independent random variables with a common continuous distribution FF, 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 πn∗ that is optimal in the sense that it maximizes the expected value of Ln(πn∗), the number of selected observations. We investigate the distribution of Ln(πn∗); in particular, we obtain a central limit theorem for Ln(πn∗) and a detailed understanding of its mean and variance for large nn. 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 NN where NN is a Poisson random variable that is independent of the sequence.