Regarding stopping rules for Brownian motion and random walks
LeRoy H. Walker · Bulletin of the American Mathematical Society · 1969
At the Fifth Berkeley Symposium on Mathematical Statistics and Probability, A. Dvoretzky [4] presented a paper on "Certain Optimal Stopping Rules."In this paper he proved the existence of an optimal stopping rule for the following situation: let Xi, X2, Xz, • • • be a sequence of independent, identically distributed, real-valued random variables with zero means and unit variances defined on the probability space (Qi, (Fi, Pi).Then for (fixed) j8> J, there exists a positive, integer-valued random variable n (called a stopping rule) defined on Qi, whose value for any sample point cois a function only of the observable variables Xi(co), ^(co), • • • , X»( W )(co) and which realizes the maximum value of £{»-*£«:!* X^ Xz, • • • is as suggested above, 1 Complete proofs of all statements made in this announcement can be found in the author's Ph.D. dissertation, Stopping rules for Brownian motion and random walks, University of California at Los Angeles, 1968