Stein’s method and the distribution of the product of zero mean correlated normal random variables

Abstract

Over the last 80 years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein’s method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.

Keywords:

• Product of correlated normal random variables
• probability density function
• Stein’s method

AMS 2010 Subject Classification:

• Primary 60E05
• 62E15

Acknowledgments

The author is supported by a Dame Kathleen Ollerenshaw Research Fellowship. The author would like to thank the referees for their helpful comments.