Journal of Applied Probability

Weak convergence to the Student and Laplace distributions

Christian Schluter and Mark Trede

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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.

Article information

J. Appl. Probab., Volume 53, Number 1 (2016), 121-129.

First available in Project Euclid: 8 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 91B30: Risk theory, insurance 91B70: Stochastic models

Limit theorem city-size growth high-frequency return process


Schluter, Christian; Trede, Mark. Weak convergence to the Student and Laplace distributions. J. Appl. Probab. 53 (2016), no. 1, 121--129.

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