Probabilities of competing binomial random variables

TL;DR

This paper analyzes the probability that one binomial random variable exceeds another by a certain margin, revealing phase transitions and properties as parameters vary, using Fourier analysis techniques.

Contribution

It introduces new asymptotic and monotonicity results for competing binomial probabilities, highlighting phase transition phenomena with integral Fourier analysis methods.

Findings

Identifies phase transition phenomena in competing binomial probabilities.

Establishes asymptotic behaviors as parameters vary.

Demonstrates monotonicity and convexity properties of the probability function.

Abstract

Suppose you and your friend both do $n$ tosses of an unfair coin with probability of heads equal to $\alpha$. What is the behavior of the probability that you obtain at least $d$ more heads than your friend if you make $r$ additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of $n$, and demonstrate surprising phase transition phenomenons as parameters $ d, r$ and $\alpha$ vary. Our main tools are integral representations based on Fourier analysis.