| Abstract: | Abstract Let X 1, X 2… and B 1, B 2… be mutually independent 0, 1]-valued random variables, with EB j = β > 0 for all j. Let Y j = B 1 … sB j?1 X j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y 1,…,Y n ):=sup{EY τ:τ is a stopping rule for Y 1,…,Y n } and E(max 1≤j≤n Y j ). It is shown that the set of ordered pairs {(x, y):x = V(Y 1,…,Y n ), y = E(max 1≤j≤n Y j ) for some sequence Y 1,…,Y n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ n, β(x)}, where Ψ n, β(x) = (1 ? β)n + 2β]x ? β?(n?2) x 2 if x ≤ β n?1, and Ψ n, β(x) = min j≥1{(1 ? β)jx + β j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained. |
