Prohorov-Type Local Limit Theorems on Abstract Wiener Spaces

Abstract

We prove that the density of $\frac{X_1+\cdots +X_n-nE[X_1]}{\sqrt{n}}$, where $\{X_n\}_{n\ge 1}$ is a sequence of independent and identically distributed random variables taking values on a abstract Wiener space, converges in $\mathcal {L}^1$ 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 $\{X_n\}_{n\ge 1}$ 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 Lanconelli and Stan (Bernoulli 22:2101–2112, 2016) 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.