Convergence in law for general distributions in Wiener and Wiener-Poisson spaces using Malliavin calculus and Stein's method
Abstract
In this dissertation a general framework to extend the Stein's method and the Nourdin-Peccati analysis is proposed. An upper bound (NP bound) for distances (that induce a stronger topology than the convergence in law) is provided in this general framework for the Wiener and Wiener-Poisson spaces. Given a reference random variable (supposed to be either Multivariate Normal or a general square integrable real-valued random variable), it is studied the solution of its Stein's equation, obtaining universal bounds on its partial derivatives. Then is extended the analysis of Nourdin and Peccati by bounding, in the proposed general framework, well chosen distances. This is illustrated with the Multivariate Normal case (which was worked already by several authors) and then is studied deeply the one dimensional version of this general NP bound, including convergence to general distributions such as members of the Exponential and Pearson families. Using these results, is obtained non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. This is done in both Wiener space and the more general Wiener-Poisson space. In the former, is studied conditions for convergence under several particular cases and characterize when two random variables have the same distribution. While in the latter space, is done an extension of some results already known in the Wiener space, such as the so-called "fourth moment theorem'' and the second order Poincaré inequality. Finally, will is shown several application to central and non-central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein-Uhlenbeck SDE driven by fractional Brownian motion), bilinear functionals of Gaussian subordinated fields where the underlying process is a fractional Brownian motion with Hurst parameter bigger than 1/2, Poisson functionals in the first Poisson chaos restricted to "small'' jumps (particularly fractional Lévy processes) and the product of two Ornstein-Uhlenbeck processes (one in the Wiener space and the other in the Poisson space).
Degree
Ph.D.
Advisors
Baudoin, Purdue University.
Subject Area
Applied Mathematics|Theoretical Mathematics
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