A stopping rule for choosing the best of three coins
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.
scientific article; zbMATH DE number 3888622