Sequential selection of an increasing sequence from a multidimensional random sample

Let random points X1,.,Xn be sampled in strict sequence from a continuous product distribution on Euclidean d-space. At the time Xj 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 .n1/(d+1) and give the exact value of .. This extends the .2n result by Samuels and Steele for d=1.