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Weak convergence to the Student and Laplace distributions

Published online by Cambridge University Press:  24 March 2016

Christian Schluter  and
Mark Trede
Christian Schluter
Affiliation:
School of Economics, CNRS and EHESS, Aix-Marseille Université, Centre de la Vieille Charité, 13002 Marseille, France.
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Abstract

One often observed empirical regularity is a power-law behavior of the tails of some distribution of interest. We propose a limit law for normalized random means that exhibits such heavy tails irrespective of the distribution of the underlying sampling units: the limit is a t-distribution if the random variables have finite variances. The generative scheme is then extended to encompass classic limit theorems for random sums. The resulting unifying framework has wide empirical applicability which we illustrate by considering two empirical regularities in two different fields. First, we turn to urban geography and explain why city-size growth rates are approximately t-distributed, using a model of random sector growth based on the central place theory. Second, turning to an issue in finance, we show that high-frequency stock index returns can be modeled as a generalized asymmetric Laplace process. These empirical illustrations elucidate the situations in which heavy tails can arise.

Keywords

Limit theoremcity-size growthhigh-frequency return process
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 121 - 129
Copyright
Copyright © Applied Probability Trust 2016 

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