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Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

Part of: Probability theory on algebraic and topological structures Distribution theory Basic linear algebra Limit theorems

Published online by Cambridge University Press:  19 February 2016

Marcus C. Christiansen  and
Nicola Loperfido
Marcus C. Christiansen*
Affiliation:
University of Ulm
Nicola Loperfido*
Affiliation:
Università degli Studi di Urbino Carlo Bo
*
∗ Postal address: Institute of Insurance Science, University of Ulm, 89081 Ulm, Germany. Email address: [email protected].
∗∗ Postal address: Dipartimento di Economia, Politica e Società, Università degli Studi di Urbino Carlo Bo, Via Saffi 42, 61029 Urbino, Italy.
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Abstract

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We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

Keywords

Central limit theoremCramer's conditionorder of convergenceskew normalskewnessthird-order tensor

MSC classification

Primary: 60F05: Central limit and other weak theorems 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Secondary: 15A69: Multilinear algebra, tensor products 62E17: Approximations to distributions (nonasymptotic)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 466 - 482
Copyright
© Applied Probability Trust 

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