## Bernoulli

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

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

#### 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
Revised: January 2015
First available in Project Euclid: 3 May 2016

https://projecteuclid.org/euclid.bj/1462297676

Digital Object Identifier
doi:10.3150/15-BEJ721

Mathematical Reviews number (MathSciNet)
MR3498024

Zentralblatt MATH identifier
1343.60016

#### 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

#### References

• [1] Barron, A.R. (1986). Entropy and the central limit theorem. Ann. Probab. 14 336–342.
• [2] Bentkus, V. (2003). On the dependence of the Berry–Esseen bound on dimension. J. Statist. Plann. Inference 113 385–402.
• [3] Bloznelis, M. (2002). A note on the multivariate local limit theorem. Statist. Probab. Lett. 59 227–233.
• [4] Bogachev, V.I. (1998). Gaussian Measures. Mathematical Surveys and Monographs 62. Providence, RI: Amer. Math. Soc.
• [5] Davydov, Y. (1992). A variant of an infinite-dimensional local limit theorem. Journal of Soviet Mathematics [1] 61 1853–1856.
• [6] Da Pelo, P., Lanconelli, A. and Stan, A.I. (2011). A Hölder–Young–Lieb inequality for norms of Gaussian Wick products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 375–407.
• [7] Da Pelo, P., Lanconelli, A. and Stan, A.I. (2013). An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems. Stochastic Process. Appl. 123 3183–3200.
• [8] Gnedenko, B.V. (1954). A local limit theorem for densities. Dokl. Akad. Nauk SSSR 95 5–7.
• [9] Holden, H., Øksendal, B., Ubøe, J. and Zhang, T. (2010). Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, 2nd ed. Universitext. New York: Springer.
• [10] Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129. Cambridge: Cambridge Univ. Press.
• [11] Lanconelli, A. and Sportelli, L. (2012). Wick calculus for the square of a Gaussian random variable with application to Young and hypercontractive inequalities. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 1250018, 16.
• [12] Lanconelli, A. and Stan, A.I. (2010). Some norm inequalities for Gaussian Wick products. Stoch. Anal. Appl. 28 523–539.
• [13] Lanconelli, A. and Stan, A.I. (2013). A Hölder inequality for norms of Poissonian Wick products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16, 39.
• [14] Linnik, Ju.V. (1959). An information-theoretic proof of the central limit theorem with Lindeberg conditions. Theory Probab. Appl. 4 288–299.
• [15] Nelson, E. (1973). The free Markoff field. J. Funct. Anal. 12 211–227.
• [16] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Berlin: Springer.
• [17] Prohorov, Yu.V. (1952). A local theorem for densities. Dokl. Akad. Nauk SSSR 83 797–800.
• [18] Ranga Rao, R. and Varadarajan, V.S. (1960). A limit theorem for densities. Sankhyā 22 261–266.