The Annals of Probability

Noncentral convergence of multiple integrals

Ivan Nourdin and Giovanni Peccati

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Fix ν>0, denote by G(ν/2) a Gamma random variable with parameter ν/2 and let n≥2 be a fixed even integer. Consider a sequence {Fk}k≥1 of square integrable random variables belonging to the nth Wiener chaos of a given Gaussian process and with variance converging to 2ν. As k→∞, we prove that Fk converges in distribution to 2G(ν/2)−ν if and only if E(Fk4)−12E(Fk3)→12ν2−48ν.

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Ann. Probab. Volume 37, Number 4 (2009), 1412-1426.

First available in Project Euclid: 21 July 2009

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Gaussian processes Malliavin calculus multiple stochastic integrals noncentral limit theorems weak convergence


Nourdin, Ivan; Peccati, Giovanni. Noncentral convergence of multiple integrals. Ann. Probab. 37 (2009), no. 4, 1412--1426. doi:10.1214/08-AOP435.

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