Sequential selection of an increasing sequence from a multidimensional random sample

Abstract

Let random points $X_1,\dots, X_n$ be sampled in strict sequence from a continuous product distribution on Euclidean $d$-space. At the time $X_j$ is observed it must be accepted or rejected. The subsequence of accepted points must increase in each coordinate. We show that the maximum expected length of a subsequence selected is asymptotic to $\gamma n^{1/(d+1)}$ and give the exact value of $\gamma$. This extends the $\sqrt{2n}$ result by Samuels and Steele for $d = 1$.