Abstract
The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of supn(Xn − cn) and supn(Sn − cn) where X1, X2, … are i.i.d. random variables, Sn = X1 + … + Xn, and cn = (nL(n))1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.
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