Beyond Wald's Equation and the Optional Sampling Theorem
Michael J. Klass, Victor H. de la Pena
TL;DR
The paper addresses stopping-time asymptotics for mean-zero martingales when the stopping time $T$ may be infinite with positive probability, extending Wald's equation and the Optional Sampling Theorem beyond traditional assumptions. It proves two main results: (i) if $E|M_T|\,\mathbf{1}(T<\infty) < \infty$, then there exists a finite $L$ with $E[M_T\,\mathbf{1}(T<\infty)] = L$ and $\lim_{n\to\infty} E[M_n\,\mathbf{1}(T>n)] = -L$, and (ii) a set of sufficient conditions ensuring $T$ is finite almost surely, along with corollaries and applications. The work also introduces three families of summands yielding computable $L$ and constant-overshoot phenomena, plus a general framework to compute $L$ for broader martingales, complemented by counterexamples illustrating the necessity of the conditions. Additionally, polynomial martingales are discussed, showing the applicability of the theory to square- and higher-degree martingale constructions, with implications for sequential analysis and related probabilistic fields.
Abstract
From the perspective of expectations of randomly stopped sums, Wald's equation and the Optional Sampling Theorem identify situations in which the stopping time can be decoupled from the stopping place, acting as if the two were independent. Herein we consider arbitrary mean zero Martingales and their extended-valued stopping times, proving two fundamental theorems and a general corollary. Examples, counter-examples and discussion are also included. The first theorem establishes conditions under which a stopped martingale's expected value can be characterized by a finite real number, also yielding a new expectation limit. Two corollaries and an application provide information on the rate of decay of the tail probability of the stopping time. The second theorem provides sufficient conditions to ensure that an extended-valued stopping time is finite with probability one. We demonstrate these results through various examples and explore their implications for different families of martingales. Our findings extend classical results in martingale theory and provide new insights into the behavior of stopped martingales, especially when the expected values are the stopped Martingale on the set where the extended-valued stopping time $T$ is finite is different from the expected value of the Martingale at time 1.