 An invariance principle under the total variation distanceIvan Nourdin and Guillaume Poly Stochastic Processes and their Applications, 2015, vol. 125, issue 6, 2190-2205 Abstract: Let X1,X2,… be a sequence of i.i.d. random variables, with mean zero and variance one and let Sn=(X1+⋯+Xn)/n. An old and celebrated result of Prohorov (1952) asserts that Sn converges in total variation to the standard Gaussian distribution if and only if Sn0 has an absolutely continuous component for some integer n0≥1. In the present paper, we give yet another proof of Prohorov’s Theorem, but, most importantly, we extend it to a more general situation. Indeed, instead of merely Sn, we consider a sequence of homogeneous polynomials in the Xi. More precisely, we exhibit conditions under which some nonlinear invariance principle, discovered by Rotar (1979) and revisited by Mossel et al. (2010), holds in the total variation topology. There are many works about CLT under various metrics in the literature, but the present one seems to be the first attempt to deal with homogeneous polynomials in the Xi with degree strictly greater than one. Keywords: Convergence in law; Convergence in total variation; Absolute continuity; Invariance principle (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:125:y:2015:i:6:p:2190-2205 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2014.12.010 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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