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Right-tail asymptotics for products of independent normal random variables

Džiugas Chvoinikov, Jonas Šiaulys

Abstract

Let $X_1,\dots,X_n$ be independent normal random variables with $X_i\sim N(μ_i,σ_i^2)$, and set $Z=\prod_{i=1}^n X_i$. We derive asymptotic approximations for the right tail probability $\mathbb{P}(Z>x)$ as $x\to\infty$. When at least one mean is nonzero, the asymptotic formula remains explicit and involves a finite multiplicative factor arising from admissible sign patterns (reflecting the different ways the product can be positive); it holds with relative error $1+O(x^{-1/n})$. The proof uses a boundary saddle-point/Laplace method: first a multidimensional Laplace approximation near the boundary saddle, then a one-dimensional endpoint Laplace approximation.

Right-tail asymptotics for products of independent normal random variables

Abstract

Let be independent normal random variables with , and set . We derive asymptotic approximations for the right tail probability as . When at least one mean is nonzero, the asymptotic formula remains explicit and involves a finite multiplicative factor arising from admissible sign patterns (reflecting the different ways the product can be positive); it holds with relative error . The proof uses a boundary saddle-point/Laplace method: first a multidimensional Laplace approximation near the boundary saddle, then a one-dimensional endpoint Laplace approximation.
Paper Structure (36 sections, 5 theorems, 177 equations)

This paper contains 36 sections, 5 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Main results
  4. Tools and lemmas
  5. General n-case: setup and tail integral
  6. Geometry of the constraint: which sign patterns matter
  7. Regime decomposition: the unbalanced regions are negligible
  8. Balanced vs unbalanced sets.
  9. Step 1: upper bound for $I_2(x)$.
  10. Step 2: lower bound for $I_1(x)$.
  11. Step 2a: control $\Psi$ on $S_+(x)$.
  12. Step 2b: integrate the lower bound.
  13. Conclusion.
  14. Saddle system
  15. Reduction via the common scalar $\lambda x$.
  16. ...and 21 more sections

Key Result

Theorem 1

Assume that at least one $\mu_i$ is nonzero. Define Then as $x\to\infty$,

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1: Computing $L_*$ and $m_*$
  • Lemma 1: Laplace approximation in $\mathbb{R}^n$
  • Corollary 1
  • proof
  • Lemma 2: Endpoint Laplace expansion at a boundary minimum
  • proof
  • Corollary 2: Endpoint rule for $\mu=\alpha=1$ with rate
  • proof