Normal Convergence Using Malliavin Calculus With Applications and Examples

TL;DR

This paper extends the Malliavin calculus framework to the Wiener-Poisson space, deriving bounds and inequalities for normal convergence, with applications to various stochastic processes and central limit theorems.

Contribution

It generalizes the chain rule and Nourdin-Peccati bound to the Wiener-Poisson space, providing new tools for analyzing normal convergence in complex stochastic settings.

Findings

Derived a second-order Poincare inequality for Wiener-Poisson space

Provided bounds for convergence rates to normality in multiple stochastic models

Unified results across Wiener, Poisson, and Wiener-Poisson spaces

Abstract

We prove the chain rule in the more general framework of the Wiener-Poisson space, allowing us to obtain the so-called Nourdin-Peccati bound. From this bound we obtain a second-order Poincare-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener-Poisson space. We also give several applications to 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), Poisson functionals in the first Poisson chaos restricted to infinitely many \small" jumps (particularly fractional Levy processes) and the product of two Ornstein-Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.