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We consider the problem of selecting sequentially a unimodal subsequence froma sequence of independent identically distributed random variables, and we findthat a person doing optimal sequential selection does within a factor of thesquare root of two as well as a prophet who knows all of the randomobservations in advance of any selections. Our analysis applies in fact toselections of subsequences that have d+1 monotone blocks, and, by including thecase d=0, our analysis also covers monotone subsequences.

Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence