A stopping rule for choosing the best of three coins
type: Item
instance of: scholarly article
title: A stopping rule for choosing the best of three coins
zbMATH Open document ID: 0557.60032
DOI: 10.1016/0304-4149(85)90047-X
author: Steven P. Lalley
published in: Stochastic Processes and their Applications
publication date: 1985-01-01T00:00:00Z
review text: Let \(\{X^ i_ n: n=1,2,...,i=1,2,3\}\) be i.i.d. random variables such that \(P(X^ i_ n=1)=p=1-P(X^ i_ n=0)\). Given \(a>0\), the following ''random times'' are considered: \(\tau_ a=the\) smallest n such that the difference between one of the sums \(\sum^{n}_{m=1}X^ i_ m\) and any of the remaining two (i.e. \(\sum^{n}_{m=1}X^ j_ m\) for \(j\neq i)\) is at least a; \(M_ a=the\) smallest \(n\geq a\) such that the difference between one of the sums \(\sum^{n}_{m=1}X^ i_ m\) and one of the remaining two is at least a; suppose that \(\sum^{M_ a}_{m=1}X^ i_ m\) is the smallest of the three sums, then \(\rho_ a\) is defined as \(-M_ a\) \(+\) the smallest \(n\geq M_ a\) such that the difference between the remaining two sums is at least a. The purpose of the paper is to evaluate \(E\tau_ a\), E \(M_ a\), \(E\rho_ a\). The random variables \(\tau_ a\), \(M_ a\) are shown to be the first passage times for a two-dimensional random walk, and asymptotic formulas (as \(a\to \infty)\) for the expectations are obtained with the help of the Wiener process in \(R^ 2\). Moreover, an algebraic (difference) equation for \(E\tau_ a\) is derived and uniqueness of its solution is discussed.
reviewed by: Tomasz Bojdecki
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3888622
zbMATH Keywords: first passage times
MaRDI profile type: Publication
cites work: Brownian motion and analytic functions
cites work: Q3241504
cites work: Q5512342
cites work: Q5815016
cites work: Selecting the highest probability in binomial or multinomial trials
description: scientific article; zbMATH DE number 3888622