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

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
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