Prohorov-Type Local Limit Theorems on Abstract Wiener Spaces

Abstract

We prove that the density of X 1 +···+Xn−nE[X 1 ] √ n , where {X n } n≥1 is a sequence of independent and identically distributed random variables taking values on a abstract Wiener space, converges in L 1 to the density of a certain Gaussian measure which is absolutely continuous with respect to the reference Wiener measure. The crucial feature in our investigation is that we do not require the covariance structure of {X n } n≥1 to coincide with the one of the Wiener measure. This produces a non trivial (different from the constant function one) limiting object which reflects the different covariance structures involved. The present paper generalizes the results proved in [18] and deepens the connection between local limit theorems on (infinite dimensional) Gaussian spaces and some key tools from the Analysis on the Wiener space, like the Wiener-Itô chaos decomposition, Ornstein-Uhlenbeck semigroup and Wick product. We also verify and discuss our main assumptions on some examples arising from the applications: dimension independent Berry-Esseen-type bounds and weak solutions of stochastic differential equations.Assumption 1.1 The law of the X n 's is absolutely continuous with respect to µ with a density f belonging to L 2 (W, µ) Assumption 1.1 has several important implications. First of all, it yields the finiteness of all the moments of the scalar random variable X n , ϕ for ϕ ∈ W * . In fact, for any m ∈ N and ϕ ∈ W * a simple application of the Cauchy-Schwartz inequality gives

Standardizing densities on Gaussian spaces

“…The role of the Wick product, and in particular of the Wick exponentials, will appear to be very natural in our approach to the above mentioned problem. Probabilistic interpretations of the Wick product have been already investigated in the papers [3], [16] and [12] for Gaussian measures, in [13] and [15] for the Poisson distribution and in [14] for the chi-square distribution. The paper is organized as follows: Section 2 is a quick overview on basic definitions and notations from the analysis on abstract Wiener spaces.…”

“…In the papers [16] and [12] the authors investigated the validity of local limit theorems, i.e. central limit theorems for densities, for a class of random variables including the example described above.…”

“…In both papers the limiting measure is the Wiener measure whose mean is zero and the covariance is the identity operator. In [16] the random variables are assumed to be already standardized while in [12] only the assumption of the first moment is relaxed. The aim of this note is to analyze the behaviour of the densities, with respect to the Wiener measaure, of random variables when we try to perturb their first and second moments.…”