Computing the covariance of two Brownian area integrals

Abstract

We compute the expected product of two correlated Brownian area integrals, a problem that arises in the analysis of a popular sorting algorithm. Along the way we find three different formulas for the expectation of the product of the absolute values of two standard normal random variables with correlation θ. These two formulas are found: (a) via conditioning and the non‐central chi‐square distribution; (b) via Mehler’s formula; (c) by representing the correlated normal random variables in terms of independent normal’s and integration using polar coordinates.