Gaussian integrals involving absolute value functions. (English) Zbl 1243.60043

Li, Wenbo V.; Wei, Ang

Houdré, Christian (ed.) et al., High dimensional probability. V: The Luminy volume. Most papers based on the presentations at the conference (HDP V), Luminy, France, May 26–30, 2008. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-78-2). Institute of Mathematical Statistics Collections 5, 43-59 (2009).

Summary: We provide general formulas to compute the expectations of absolute value and sign of Gaussian quadratic forms, i.e. \(\mathbb{E} |\langle X, A\)X\(\rangle +\langle b\), \(X\rangle +c|\) and \(\mathbb{E}\mathrm{sgn}(\langle X, AX\rangle +\langle b, X\rangle +c)\) 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.
For the entire collection see [Zbl 1228.60006].

MSC:

60G50	Sums of independent random variables; random walks
60E15	Inequalities; stochastic orderings