Quantitative stable limit theorems on the Wiener space

Abstract

We use Malliavin operators in order to prove quantitative stable limit theorems on the Wiener space, where the target distribution is given by a possibly multidimensional mixture of Gaussian distributions. Our findings refine and generalize previous works by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39-64] and Harnett and Nualart [Stochastic Process. Appl. 122 (2012) 3460-3505], and provide a substantial contribution to a recent line of research, focussing on limit theorems on the Wiener space, obtained by means of the Malliavin calculus of variations. Applications are given to quadratic functionals and weighted quadratic variations of a fractional Brownian motion.

Published at http://dx.doi.org/10.1214/14-AOP965 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Central limit theorems for sequences of multiple stochastic integrals
• Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion
• Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model
• Central limit theorem for a Stratonovich integral with Malliavin calculus
• Modified Euler approximation scheme for stochastic differential equations driven by fractional Brownian motions

Cited by in corpus (6)

• Asymptotic expansion of a variation with anticipative weights
• Asymptotic expansion of Skorohod integrals
• Weak convergence on Wiener space: targeting the first two chaoses
• Stable limit theorems on the Poisson space
• Rate of convergence for the weighted Hermite variations of the fractional Brownian motion
• Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance