Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Win rates at first-passage times for biased simple random walks

F. Thomas Bruss, Davy Paindaveine

TL;DR

The paper analyzes how stopping a biased simple random walk at the first-passage time to level d biases the observed win-rate Rd/Nd. It derives generating-function based integral representations for the first two moments, providing explicit, parity-dependent closed forms and establishing monotonicity and asymptotic behavior as d grows. A key contribution is showing how first-passage sampling can yield unbiased estimators of constants like π when paired with biased coin flips, with biasing significantly reducing computational cost and improving accuracy. The work also discusses extensions to other processes and stopping scenarios, as well as limitations including heavy-tailed hitting times under fair-coin settings and potential avenues for future research.

Abstract

We study the win rate $R_{N_d}/N_d$ of a biased simple random walk $S_n$ on $\mathbb{Z}$ at the first-passage time $N_d=\inf\{n\ge 0:S_n=d\}$, with $p=P[X_1=+1]\in[1/2,1)$. Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of $R_{N_d}/N_d$ along with monotonicity properties in the threshold $d$ and the bias $p$. We also provide closed-form expressions and use them to design unbiased coin-flipping estimators of $π$ based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.

Win rates at first-passage times for biased simple random walks

TL;DR

The paper analyzes how stopping a biased simple random walk at the first-passage time to level d biases the observed win-rate Rd/Nd. It derives generating-function based integral representations for the first two moments, providing explicit, parity-dependent closed forms and establishing monotonicity and asymptotic behavior as d grows. A key contribution is showing how first-passage sampling can yield unbiased estimators of constants like π when paired with biased coin flips, with biasing significantly reducing computational cost and improving accuracy. The work also discusses extensions to other processes and stopping scenarios, as well as limitations including heavy-tailed hitting times under fair-coin settings and potential avenues for future research.

Abstract

We study the win rate of a biased simple random walk on at the first-passage time , with . Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of along with monotonicity properties in the threshold and the bias . We also provide closed-form expressions and use them to design unbiased coin-flipping estimators of based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.
Paper Structure (5 sections, 4 theorems, 54 equations, 2 figures)

This paper contains 5 sections, 4 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. An explicit expression for the expectation
  4. An application: fast approximation of $\pi$ with a coin
  5. Extensions and limitations

Key Result

Theorem 2.1

Fix $p\in[\frac{1}{2},1)$ and a positive integer $d$. Then, and where we let

Figures (2)

  • Figure 1: Plots of $\log(d{\rm Var}[\hat{\pi}_d])$ and $\log(d{\rm Var}[\tilde{\pi}_d])$ versus $d\in\{1,3,\ldots,101\}$.
  • Figure 2: Monte Carlo illustrations for the unbiased estimators $\hat{\pi}_d$ (fair coin, $p=\tfrac12$) and $\tilde{\pi}_d$ (biased coin, $p=\tfrac34$), over odd thresholds $d\in\{1,3,5,7,9\}$. (Left:) for each $d$, the two shaded bands show the intervals ${\rm Ave}[\hat{\pi}_d]\pm \tfrac12\,{\rm SD}[\hat{\pi}_d]$ and ${\rm Ave}[\tilde{\pi}_d]\pm \tfrac12\,{\rm SD}[\tilde{\pi}_d]$, based on the averages and standard deviations obtained from $M=100$ replications; the horizontal line marks $\pi$. (Right:) sample medians of the corresponding hitting times $N_d$ (one curve for each estimator), with the numerical labels indicating, for each $d$, the maximum observed value of $N_d$ among the $M=100$ replications.

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • proof