Gaussian integrals involving absolute value functions

Evaluating the Gaussian integrals (expectation, moments, etc.) involving the absolute value function has been playing important roles in various contents. For example, in [KN08] and [LW09], the expected number of zeros of random harmonic functions, which is also the average number of images of certain gravitational lensing system, was associated with the expectation of absolute value of certain Gaussian quadratic forms. In [BD00], the dislocation point density of Gaussian random wave was expressed as the expectation of absolute value of certain Gaussian quadratic form. In [LMOS83], the authors were interested in the average of absolute multiplicative structures, e.g. E |X1X2 · · ·Xn|, which arised in the analysis of learning curves of many adaptive systems. Selberg’s integral and Mehta’s integral are also equivalent to this structure for certain Gaussian random vectors, see [M04]. Very recently, an elegant Gaussian inequality E |X1X2 · · ·Xn| ≤ √ perm(EXiXj), due to the first author, was established in [LW09+], where perm(EXiXj) is the permanent of the covariance matrix of the centered Gaussian vector (X1, X2, ..., Xn). The explicit expression of the simplest absolute multiplicative structure E |X1X2| and related series expansions were re-derived in [SW01], and were used to study the correlation between two dependent Brownian area integrals. Here we concentrate on exact evaluations which also appeared in the theory of Gaussian random matrices in various settings. In [AW08], the authors used the spectral analysis on the Gaussian Orthogonal Ensemble random matrix to compute the first order approximation for stationary isotropic process defined on a polyhedron, which provided an upper bound for the density of maximum of certain smooth Gaussian fields. In particular, they dealt with the expectation E |det(Gn − νIn)| where Gn was a GOE matrix and In standed for the n × n identity matrix. A Gaussian representation for the intrinsic volumes of convex body was given in terms of E |detM |, where M was the random matrix with independent standard Gaussian entries, see [V08]. In [DSZ06], a special Gaussian integral involving absolute value function was studied to provide the density of critical points of given holomorphic section which was related to counting vacua in string theory. In this article, we also provide a formula for the expected sign of Gaussian quadratic forms, which is also useful in applications. For example, the best known constant in Grothendieck inequality was obtained by using the expectation E sgn (XY ) where X and Y are Gaussians, see [K79]. In [W75], the author also used this expectation to study the proportion of the time that a Brownian sheet on [0, 1] is positive. The explicit expression of E sgn (XY ), see Corollary 3.1, is often known as Sheppard’s formula; see e.g. [BD76]. In general, evaluating Gaussian integrals involving absolute value functions or sign functions are technically difficult, and there is no universal method available. In this article, we provide a systematic study of techniques and associated examples. In particular, we focus on Gaussian quadratic forms. This paper is organized as follows: Section 2 is about the representations of absolute function and sign function which are helpful in dealing with quadratic forms of Gaussian random variables. Several interesting corollaries and examples are included in Section 3 based on these representations. Most of these results are new and of independent interest. In Section 4, we discuss other approaches for Gaussian integrals involving absolute value functions.