Orderings of optimal stopping values and prophet inequalities for certain multivariate distributions (Q756264)

For any finite sequence \(X=(X_ 1,X_ 2,...,X_ n)\) of random variables, \(V(X)=\sup_{t}E(X_ t)\) is the optimal return for all stopping times t. This paper investigates orderings of such sequences under which V(X) is monotone. First, it is proved that if \(Z_ 1,Z_ 2,...,Z_{n+1}\) are independent, \(PF_ 2\) random variables, and if \(X^{(s)}=(X_ 1^{(s)},X_ 2^{(s)},...,X_ n^{(s)})\) is the n-dimensional vector having the conditional distribution of \((Z_ 1,Z_ 2,...,Z_ n)\), given that \(\sum^{n+1}_{k=1}Z_ k=s\), then \(s<s^*\) implies \(V(X^{(s)})\leq V(X^{(s^*)}).\) Finally, sequences \((X_ 1,X_ 2,...,X_ n)\) obtained by random sampling with partial replacement (where the k-th observation is replaced with probability \(\pi_ k)\) are considered. It is proved that sampling without replacement \((\pi_ k=0\) for \(k=1,2,...,n)\) always yields a higher value V than any partial replacement, but beyond this, no monotonicity results are given; instead, the prophet inequality is proved in this context, and some open problems are discussed.