 
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
Pn 1
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
