Two properties of vectors of quadratic forms in Gaussian random variables

Vladimir I. Bogachev (1) , Egor D. Kosov (1) , Ivan Nourdin (2) , Guillaume Poly (3)

1 Faculty of Mechanics and Mathematics [Moscow]
2 IECL - Institut Élie Cartan de Lorraine
3 FSTC - Faculté des Sciences, de la Technologie et de la Communication

We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting consequences. Our second result gives a characterization of limits in law for sequences of such vectors.

Mots clés: absolute continuity; convergence in law; second Wiener chaos; Quadratic forms

Domaines: Probabilités [math.PR]

Vladimir I. Bogachev, Egor D. Kosov, Ivan Nourdin, Guillaume Poly. Two properties of vectors of quadratic forms in Gaussian random variables. Theory of Probability and Its Applications c/c of Teoriia Veroiatnostei i Ee Primenenie, 2014, 59 (2), pp.208-221. ⟨10.1137/S0040585X97T987041⟩. ⟨hal-00826022⟩