On the validity of Wald's equation

Abstract

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X1 to the case where the expectation of X1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.