Weak convergence on Wiener space: targeting the first two chaoses

@article{Krein2019WeakCO,
  title={Weak convergence on Wiener space: targeting the first two chaoses},
  author={Christian Yves L{\'e}opold Krein},
  journal={Latin American Journal of Probability and Mathematical Statistics},
  year={2019}
}
  • C. Krein
  • Published 2019
  • Mathematics
  • Latin American Journal of Probability and Mathematical Statistics
We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Gamma-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence… Expand
A bound on the Wasserstein-2 distance between linear combinations of independent random variables
Abstract We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of l 2 ( N ∗ ) . We use this bound to estimate the Wasserstein-2 distance betweenExpand
Malliavin-Stein Method: a Survey of Recent Developments
Initiated around the year 2007, the Malliavin-Stein approach to probabilistic approximations combines Stein's method with infinite-dimensional integration by parts formulae based on the use ofExpand
Malliavin–Stein method: a survey of some recent developments

References

SHOWING 1-10 OF 26 REFERENCES
Quantitative stable limit theorems on the Wiener space
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 GaussianExpand
Stable Convergence of Multiple Wiener-Itô Integrals
We prove sufficient conditions ensuring that a sequence of multiple Wiener-Itô integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Note thatExpand
Convergence in law in the second Wiener/Wigner chaos
Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset $L_0$ of $L$ satisfying that, for any $F_\infty$ in $L_0$, the convergence of onlyExpand
Stein’s method on Wiener chaos
We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general GaussianExpand
Convergence in total variation on Wiener chaos
Let {Fn} be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F∞ satisfying V ar(F∞)>0. Our first result is aExpand
Central limit theorems for multiple stochastic integrals and Malliavin calculus
We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of theExpand
Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach
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,Expand
Noncentral convergence of multiple integrals
Fix ν>0, denote by G(v/2) a Gamma random variable with parameter v/2, and let n≥2 be a fixed even integer. Consider a sequence (F_k) of square integrable random variables, belonging to the nth WienerExpand
Asymptotic independence of multiple Wiener-It\^o integrals and the resulting limit laws
We characterize the asymptotic independence between blocks consisting of multiple Wiener-It\^{o} integrals. As a consequence of this characterization, we derive the celebrated fourth moment theoremExpand
Non-Universality of Nodal Length Distribution for Arithmetic Random Waves
Abstract“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur etExpand
...
1
2
3
...