On stopping rules and the expected supremum of S_n∕T_n

Michael Jay Klass and Lawrence Edward Myers

Vol. 75 (1978), No. 2, 467–476

DOI: 10.2140/pjm.1978.75.467

Abstract

Let S_n and T_n 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 ℬ_n be the σ-field generated by S₁,T₁,⋯,S_n,T_n, and let R_∞ be the collection of extended-valued stopping rules with respect to ℬ₁,ℬ₂,⋯ . It is shown that E sup_n≧1S_n∕T_n < ∞ ifi sup_τ∈R∞ES_τ∕T_τ < ∞ iff ES₁ log ⁺S₁ < ∞ and E(T₁⁻¹) < ∞. 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 S_n∕T_n framework.

Mathematical Subject Classification 2000

Primary: 60G40

Secondary: 62L15

Milestones

Received: 20 April 1976

Published: 1 April 1978