Oscillation criteria for stopping near the top of a random walk

TL;DR

This paper investigates the stopping problem in Bernoulli random walks, focusing on the asymptotic behavior of an optimal strategy sequence and providing bounds and properties to approach a conjectured limit.

Contribution

It establishes the best lower bound for the sequence determining the optimal stopping strategy and explores its properties to address the conjecture about its asymptotic behavior.

Findings

Established a new lower bound for the sequence p_{n} that determines the optimal stopping rule.

Proved additional properties of the sequence p_{n} to support the conjecture that p_{n} approaches 1/2.

Progressed towards confirming Allaart's conjecture on the asymptotic behavior of the sequence.

Abstract

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.