José A. Islas | Searcharxiv

arXiv2017

Oscillation criteria for stopping near the top of a random walk

Consider the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter $p$ and finite time horizon $n$. Allaart \cite{Allaart} proved that the optimal strategy is determined by an interesting sequence of constants $\{p_{n}\}$. He conjectured the asymptotic behavior to be $1/2$. In this work the best lower bound for this sequence is found and more of its properties are proven towards solving the conjecture.

math.PR↗

arXiv2017

A sharp bound for winning within a proportion of the maximum of a sequence

This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let $M_n:=\max\{X_1,\dots,X_n\}$, and the objective is to select the maximum of the sequence. What is the maximum probability of "stopping at the maximum"? That is, what is the stopping time $τ$ adapted to $X_1,...,X_n$ that maximizes $P(X_τ=M_n)$? This problem was examined by Gilbert and Mosteller \cite{GilMost} when in addition the common distribution is continuous. The optimal win probability in this case is denoted by $v_{n,max}^*$. What if it is desired to "stop within a proportion of the maximum"? That is, for $0<α<1$, what is the stopping rule $τ$ that maximizes $P(X_τ \geq αM_n)$? In this note both problems are treated as games, it is proven that for any continuous random variable $X$, if $τ^*$ is the optimal stopping rule then $P(X_{τ^*} \geq αM_n)\geq v_{n,max}^*$, and this lower bound is sharp. Some examples and another interesting result are presented.

math.PR↗