Vol. 75, No. 2, 1978

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On stopping rules and the expected supremum of Sn∕Tn

Michael Jay Klass and Lawrence Edward Myers

Vol. 75 (1978), No. 2, 467–476
Abstract

Let Sn and Tn be n-th partial sums of two independent sequences of i.i.d. random variables. S1 and T1 may have different distributions. Assume 0 ES1 < , ET1 < and P[T1 > 0] = 1. Let n be the σ-field generated by S1,T1,,Sn,Tn, and let R be the collection of extended-valued stopping rules with respect to 1,2, . It is shown that E supn1Sn∕Tn < ifi supτRESτ∕Tτ < iff ES1 log +S1 < and E(T11) < . The (random) cutoff points characterizing the optimal rules are easily obtained as fixed points of certain contraction mappings. A Markov walk generalization of the Chow and Robbins binomial stopping problem is viewed within the Sn∕Tn framework.

Mathematical Subject Classification 2000
Primary: 60G40
Secondary: 62L15
Milestones
Received: 20 April 1976
Published: 1 April 1978
Authors
Michael Jay Klass
Lawrence Edward Myers