A Special Distributional Result for Bilinear Forms

Abstract

Necessary and sufficient conditions are given such that a quadratic form has moment-generating function E[exp ( tU′BU)] = (1 – t ²)^–r/4 for |t| < 1 with U ∼ N_k (μ, Σ) and Σ positive definite. An important corollary gives conditions under which the bilinear form X′AY involving two different multivariate normal random vectors (of not necessarily the same dimensions) has the same distribution as the sum of independent random variables, each having the LaPlace (double exponential) distribution.

Key Words:

• LaPlace
• Double exponential
• Correlation
• Normal products