Gaussian integrals involving absolute value functions

Abstract

We provide general formulas to compute the expectations of absolute value and sign of Gaussian quadratic forms, i.e. E | X, AX + b, X + c| and E sgn( X, AX + b, X + c) for centered Gaussian random vector X, fixed matrix A, vector b and constant c. Products of Gaussian quadratics are also discussed and followed with several interesting applications.

Spatial Bunching of Same-Index Polarization Singularities in Two-Dimensional Random Vector Waves

“…In some particular cases [26,37,52], it is possible to derive a closed analytical expression for averages of the form in Eq. (16) by reducing the integrand to a quadratic form and integrating with standard mathematical techniques [54]. However, the specific form of the correlation matrix in our model does not lend itself easily to applying the formalism of Ref.…”

On the expected number of zeros of a random harmonic polynomial

“…Our method is based on a representation of the absolute function. A useful corollary is given; for related topics of independent interest, see details in [LW08].…”

Slope distribution in front-back asymmetric stochastic Lagrange time waves

“…In this paper, they are found by simulation of the four-dimensional conditional normal variables, followed by numerical integration. An alternative could be to use the RINDmodule in the WAFO toolbox (see http://www.maths.lth.se/matstat/wafo/ and also [8]). …”

“…In this example, the physically motivated linked model (5) will be compared to the forced correlation model (8). For each combination of depth h and α-parameter in the linked model, the parameters γ and δ in (8) are chosen to make the variance var(X(t, u)) and covariance cov(W (t, u), X(t, u)) equal in the two models, so the two models have identical bivariate distributions of (W (t, u), X(t, u)).…”

“…For each combination of depth h and α-parameter in the linked model, the parameters γ and δ in (8) are chosen to make the variance var(X(t, u)) and covariance cov(W (t, u), X(t, u)) equal in the two models, so the two models have identical bivariate distributions of (W (t, u), X(t, u)). It appears that even if the forced and linked models have the same (W (t, u), X(t, u))-distributions, they differ considerably with respect to slope distributions-the forced model giving greater front-back asymmetry.…”