Bernoulli

  • Bernoulli
  • Volume 22, Number 4 (2016), 2101-2112.

A note on a local limit theorem for Wiener space valued random variables

Alberto Lanconelli and Aurel I. Stan

Full-text: Open access

Abstract

We prove a local limit theorem, that is, a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein–Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired $\mathcal{L}^{1}$-convergence of the density of $\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$. We close the paper comparing our result with certain Berry–Esseen bounds for multidimensional central limit theorems.

Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 2101-2112.

Dates
Received: May 2014
Revised: January 2015
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1462297676

Digital Object Identifier
doi:10.3150/15-BEJ721

Mathematical Reviews number (MathSciNet)
MR3498024

Zentralblatt MATH identifier
1343.60016

Keywords
abstract Wiener space local limit theorem Ornstein–Uhlenbeck semigroup Wick product

Citation

Lanconelli, Alberto; Stan, Aurel I. A note on a local limit theorem for Wiener space valued random variables. Bernoulli 22 (2016), no. 4, 2101--2112. doi:10.3150/15-BEJ721. https://projecteuclid.org/euclid.bj/1462297676


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