Summary: A Sharp Ratio Inequality for Optimal Stopping When Only
Record Times are Observed
Pieter C. Allaart
Department of Mathematics, University of North Texas, Denton, Texas, USA
Abstract: Let X1; : : : ; Xn be independent, identically distributed random variables that are non-
negative and integrable, with known continuous distribution. These random variables are observed
sequentially, and the goal is to maximize the expected X value at which one stops. Let Vn denote
the optimal expected return of a player who can observe at time j only whether Xj is a relative
record (j = 1; : : : ; n), and Wn that of a player who observes at time j the actual value of Xj. It is
shown that Vn > anWn, where an = max1 k
j=k 1=j, and this inequality is sharp.
Keywords: Optimal stopping rule; Relative record; Secretary problem.
Subject Classications: 60G40.
Let X;X1;:::;Xn be independent, identically distributed (i.i.d.) random variables that
are nonnegative and integrable, with known continuous distribution. Recently, Samuel-
Cahn (2007) investigated the problem of maximizing EX over those stopping times that
use only information about the record times of the sequence X1;:::;Xn. Precisely, let
Y1 1, and for j = 2;:::;n, let Yj = 1 if Xj > maxfX1;:::;Xj 1g, and Yj = 0 otherwise.
Dene the -algebras qj := (fY1;:::;Yjg), j = 1;:::;n, and let