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

Vlad Bally 1, 2 Lucia Caramellino 3 Guillaume Poly 4
1 MATHRISK - Mathematical Risk Handling
Inria de Paris, ENPC - École des Ponts ParisTech, UPEM - Université Paris-Est Marne-la-Vallée
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 Dipartimento di Matematica [Rome]
4 IRMAR - Institut de Recherche Mathématique de Rennes
Abstract : We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables. 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 Xn, n∈N, on hand is needed. Essentially, one needs that the law of Xn is locally lower bounded by the Lebesgue measure (Doeblin's condition). This topic is also widely discussed in the literature (see the book by Battacharaya and Rao). Our main contribution is to discuss convergence in distribution norms, that is to replace the test function f by some derivative ∂αf and to obtain upper bounds for εn(∂αf) in terms of the original function f.
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Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2018, 23, paper 45, 51 p. 〈10.1214/18-EJP174〉
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Vlad Bally, Lucia Caramellino, Guillaume Poly. Convergence in distribution norms in the CLT for non identical distributed random variables. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2018, 23, paper 45, 51 p. 〈10.1214/18-EJP174〉. 〈hal-01413548〉

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