On the Gaussian Mixture Representation of the Laplace Distribution

ABSTRACT

Under certain conditions, a symmetric unimodal continuous random variable ξ can be represented as a scale mixture of a standard Normal distribution Z, that is, $\xi =\sqrt{W}Z$, where the mixing distribution W is independent of Z. It is well known that if the mixing distribution is inverse Gamma, then ξ has Student’s t distribution. However, it is less well known that if the mixing distribution is Gamma, then ξ has a Laplace distribution. Several existing proofs of the latter result rely on complex calculus or nontrivial change of variables in integrals. We offer two simple and intuitive proofs based on representation and moment generating functions. As a byproduct, our proof by representation makes connections to many existing results in statistics. Supplementary materials for this article are available online.

KEYWORDS:

• Bartlett decomposition
• Gauss–Laplace representation
• Legendre’s duplication formula
• Moment generating function
• Scale mixture

Acknowledgments

Both authors benefited from Professor Carl N. Morris’ enthusiasm for representing complicated random variables by simple ones. The authors thank the reviewers for giving very valuable comments. The authors thank Dr. Avi Feller at the Goldman School of Public Policy at Berkeley for bringing Professor Christian Robert’s blog article on Gauss–Laplace representation (https://xianblog.wordpress.com/2015/10/14/gauss-to-laplace-transmutation/) into their attention. In the blog article, Professor Robert uses the term “Gauss–Laplace transmutation.” The authors avoid using the term “transmutation” because it was used for other purposes in the literature.