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Large deviations of means of heavy-tailed random variables with finite moments of all orders

Part of: Limit theorems Nonparametric inference Distribution theory - Probability

Published online by Cambridge University Press:  04 April 2017

Jaakko Lehtomaa
Jaakko Lehtomaa*
Affiliation:
University of Helsinki
*
* Postal address: Department of Mathematics and Statistics, University of Helsinki, PO Box 68, 00014, Helsinki, Finland. Email address: jaakkolehtomaa@gmail.com
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Abstract

Logarithmic asymptotics of the mean process {Sn∕n} are investigated in the presence of heavy-tailed increments. As a consequence, a full large deviations principle for means is obtained when the hazard function of an increment is regularly varying with index α∈(0,1). This class includes all stretched exponential distributions. Thus, the previous research of Gantert et al. (2014) is extended. Furthermore, the presented proofs are more transparent than the techniques used by Nagaev (1979). In addition, the novel approach is compatible with other common classes of distributions, e.g. those of lognormal type.

Keywords

Logarithmic asymptoticsregular variationstretched exponentiallarge deviationheavy-tailed

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations 62G32: Statistics of extreme values; tail inference
Secondary: 60E05: Distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 66 - 81
Copyright
Copyright © Applied Probability Trust 2017 

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