Electronic Journal of Probability

Convergence in distribution norms in the CLT for non identical distributed random variables

Vlad Bally, Lucia Caramellino, and Guillaume Poly

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We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is \[ \varepsilon _{n}(f):={\mathbb{E} }\Big (f\Big (\frac 1{\sqrt n}\sum _{i=1}^{n}Z_{i}\Big )\Big )-{\mathbb{E} }\big (f(G)\big )\rightarrow 0 \] where $Z_{i}$, $i\in \mathbb{N} $, are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables $Z_{i}$, $i\in{\mathbb {N}} $, on hand is needed. Essentially, one needs that the law of $Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin’s condition). This topic is also widely discussed in the literature. Our main contribution is to discuss convergence in distribution norms, that is to replace the test function $f$ by some derivative $\partial _{\alpha }f$ and to obtain upper bounds for $\varepsilon _{n}(\partial _{\alpha }f)$ in terms of the infinite norm of $f$. Some applications are also discussed: an invariance principle for the occupation time for random walks, small balls estimates and expected value of the number of roots of trigonometric polynomials with random coefficients.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 45, 51 pp.

Received: 17 October 2017
Accepted: 28 April 2018
First available in Project Euclid: 25 May 2018

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Primary: 60F05: Central limit and other weak theorems 60B10: Convergence of probability measures 60H07: Stochastic calculus of variations and the Malliavin calculus

central limit theorems abstract Malliavin calculus integration by parts regularizing results

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Bally, Vlad; Caramellino, Lucia; Poly, Guillaume. Convergence in distribution norms in the CLT for non identical distributed random variables. Electron. J. Probab. 23 (2018), paper no. 45, 51 pp. doi:10.1214/18-EJP174. https://projecteuclid.org/euclid.ejp/1527213726

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