Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach

Ehsan Azmoodeh (1) , Giovanni Peccati (2) , Guillaume Poly (3, 2)

1 Mathematics Research Unit
2 FSTC - Faculté des Sciences, de la Technologie et de la Communication
3 IRMAR - Institut de Recherche Mathématique de Rennes

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators $\Gamma_i$ (introduced by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly \cite{n-po-1}, concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

Mots clés: weak convergence; multiple integrals; cumulants; noncentral limit theorems; Malliavin calculus; Gaussian analysis

Domaines: Probabilités [math.PR]

Ehsan Azmoodeh, Giovanni Peccati, Guillaume Poly. Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach. Catherine Donati-Martin; Antoine Lejay; Alain Rouault. In Memoriam Marc Yor : Séminaire de Probabilités XLVII, 2137, Springer, pp.339-367, 2015, Lecture Notes in Mathematics, 978-3-319-18584-2. ⟨10.1007/978-3-319-18585-9_16⟩. ⟨hal-01110481⟩