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Mathematics > Probability

arXiv:1812.08268 (math)
[Submitted on 19 Dec 2018 (v1), last revised 2 Jan 2019 (this version, v2)]

Title:A multivariate central limit theorem for Lipschitz and smooth test functions

Authors:Martin Raič
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Abstract:We provide an abstract multivariate central limit theorem with the Lindeberg-type error bounded in terms of Lipschitz functions (Wasserstein 1-distance) or functions with bounded second or third derivatives. The result is proved by means of Stein's method. For sums of i.i.d. random vectors with finite third absolute moment, the optimal rate of convergence is established (that is, we eliminate the logarithmic factor in the case of Lipschitz test functions). We indicate how the result could be applied to certain other dependence structures, but do not derive bounds explicitly.
Comments: 26 pages
Subjects: Probability (math.PR)
MSC classes: 60F05
Cite as: arXiv:1812.08268 [math.PR]
  (or arXiv:1812.08268v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1812.08268

Submission history

From: Martin Raič [view email]
[v1] Wed, 19 Dec 2018 22:09:12 UTC (29 KB)
[v2] Wed, 2 Jan 2019 10:00:14 UTC (30 KB)
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