Limiting distributions of ratios of Binomial random variables

TL;DR

This paper investigates the limiting distribution of ratios of independent Binomial variables raised to powers, showing convergence to a Normal distribution under various conditions, supported by simulations.

Contribution

It provides new theoretical results on the asymptotic distribution of ratios of Binomial variables with different parameters, including explicit convergence conditions.

Findings

Ratios converge to a Normal distribution under certain conditions

Theoretical results are validated through simulations

Provides explicit mean and variance formulas for the limiting distribution

Abstract

We consider the limiting distribution of the quantity $X^s/(X+Y)^r$, where $X$ and $Y$ are two independent Binomial random variables with a common success probability and a number of trials $n$ and $m$, respectively, and $r,s$ are positive real numbers. Under several settings, we prove that this converges to a Normal distribution with a given mean and variance, and demonstrate these theoretical results through simulations.