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A Dynamic Taylor’s law

Part of: Stochastic processes Applications Limit theorems

Published online by Cambridge University Press:  24 June 2022

Victor De La Pena ,
Paul Doukhan  and
Yahia Salhi Open the ORCID record for Yahia Salhi [Opens in a new window]
Victor De La Pena*
Affiliation:
Columbia University
Paul Doukhan*
Affiliation:
CY University
Yahia Salhi*
Affiliation:
University of Lyon
*
*Postal address: Columbia University, New York, NY 10027, USA. Email address: [email protected]
**Postal address: CY University, AGM UMR 8088, site Saint-Martin, 95000 Cergy-Pontoise, France. Email address: [email protected]
***Postal address: University of Lyon, UCBL, ISFA LSAF EA2429, 69007 Lyon, France. Email address: [email protected]
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Abstract

Taylor’s power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. The law has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics, and mathematics. In particular, it has been observed in problems involving population dynamics, market trading, thermodynamics, and number theory. In applications, many authors consider panel data in order to obtain laws of large numbers. Essentially, we aim to consider ergodic behaviors without independence. We restrict our study to stationary time series, and develop different Taylor exponents in this setting. From a theoretical point of view, there has been a growing interest in the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor’s law related to independent samples. In this paper we introduce a dynamic Taylor’s law for dependent samples using self-normalized expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of the estimation involves a series of covariances, unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for a goodness-of-fit procedure suitable for checking whether the corresponding dynamic Taylor’s law holds in empirical studies.

Keywords

Self-normalized sumsTaylor’s lawweak dependencecentral limit theorem

MSC classification

Primary: 60G99: None of the above, but in this section 60F05: Central limit and other weak theorems
Secondary: 62P12: Applications to environmental and related topics
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 584 - 607
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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