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The distribution of the ratio of products of independent zero mean normal random variables

Robert E. Gaunt, Heather L. Sutcliffe

TL;DR

This work derives the exact distribution of the ratio $Z=X/Y$ where $X$ and $Y$ are products of independent zero-mean normal variables with variances $\sigma_{X_i}^2$ and $\sigma_{Y_j}^2$, respectively. It provides closed-form expressions for the PDF, CDF, and characteristic function of $Z$ in terms of the Meijer $G$-function for general $(M,N)$, along with fractional moments and asymptotic behavior. The authors also analyze asymptotics, tail behavior, and quantile functions, and explore numerous special cases (including products of Cauchy variables and reciprocal normals) that yield simplifications or known functions. These results unify and extend distributional properties of normal-product ratios, enabling exact and asymptotic calculations of tails, quantiles, and moments for a broad class of models with product-ratio structure.

Abstract

Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $σ_{X_i}^2$, $i=1,\ldots,M$, and $σ_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$. In this paper, we derive the exact probability density function of the ratio $X/Y$. We apply this formula to derive exact formulas for the cumulative distribution function and the characteristic function. We also obtain further distributional properties, including asymptotic approximations for the probability density function, tail probabilities and the quantile function.

The distribution of the ratio of products of independent zero mean normal random variables

TL;DR

This work derives the exact distribution of the ratio where and are products of independent zero-mean normal variables with variances and , respectively. It provides closed-form expressions for the PDF, CDF, and characteristic function of in terms of the Meijer -function for general , along with fractional moments and asymptotic behavior. The authors also analyze asymptotics, tail behavior, and quantile functions, and explore numerous special cases (including products of Cauchy variables and reciprocal normals) that yield simplifications or known functions. These results unify and extend distributional properties of normal-product ratios, enabling exact and asymptotic calculations of tails, quantiles, and moments for a broad class of models with product-ratio structure.

Abstract

Let and be independent zero mean normal random variables with variances , , and , , respectively, and let and . In this paper, we derive the exact probability density function of the ratio . We apply this formula to derive exact formulas for the cumulative distribution function and the characteristic function. We also obtain further distributional properties, including asymptotic approximations for the probability density function, tail probabilities and the quantile function.
Paper Structure (8 sections, 9 theorems, 41 equations)

This paper contains 8 sections, 9 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The normal product ratio distribution
  3. Probability density function
  4. Cumulative distribution function
  5. Characteristic function
  6. Asymptotic behaviour of the distribution
  7. Special cases
  8. The Meijer $G$-function

Key Result

Theorem 2.1

Suppose $M,N\geq0$ and $M+N\geq1$. Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent random variables where $X_i\sim N(0,\sigma^2_{X_i})$, $i=1,\ldots,M$, and $Y_j\sim N(0,\sigma^2_{Y_j})$, $j=1,\ldots,N$. Let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$, and denote $Z=X/Y$. Then, for $z\in\mathbb

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 10 more