A half-normal distribution scheme for generating functions

TL;DR

This paper introduces a theorem characterizing when bivariate generating functions lead to half-normal distributions, and applies it to lattice path problems and Banach's matchbox problem, extending previous results.

Contribution

It provides a general theorem linking generating functions to half-normal distributions and applies it to various combinatorial problems, broadening the understanding of their probabilistic behavior.

Findings

Number of returns to zero follows a half-normal distribution in certain lattice paths.

Height and sign changes in lattice paths are also half-normally distributed under zero drift.

The results extend known distributions to more general step sets and offer a new proof of Banach's matchbox problem.

Abstract

We present a general theorem on the structure of bivariate generating functions which gives sufficient conditions such that the limiting probability distribution is a half-normal distribution. If $X$ is a normally distributed random variable with zero mean, then $|X|$ obeys a half-normal distribution. In the second part, we apply our result to prove three natural appearances in the domain of lattice paths: the number of returns to zero, the height, and the sign changes are under zero drift distributed according to a half-normal distribution. This extends known results to a general step set. Finally, our result also gives a new proof of Banach's matchbox problem.