Limit Distribution of Two Skellam Distributions, Conditionally on Their Equality

TL;DR

This paper investigates the behavior of two Skellam-distributed variables with parameters tending to infinity, showing that conditioned on their equality, the first variable converges to a Gaussian distribution.

Contribution

It establishes the limiting distribution of one Skellam variable conditioned on equality with another, revealing Gaussian behavior in the asymptotic regime.

Findings

Conditional distribution converges to Gaussian as parameters grow large

Limit distribution depends on the parameters tending to infinity

Provides theoretical insight into Skellam distribution behavior under conditioning

Abstract

Consider two random variables following Skellam distributions of parameters going to infinity linearly. We prove that the limit distribution of the first variable, conditionally on being equal to the second, is Gaussian.