Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T01:19:01.088Z Has data issue: false hasContentIssue false
  • English
  • Français

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: vhdl@columbia.edu
**Postal address: CY University, AGM UMR 8088, site Saint-Martin, 95000 Cergy-Pontoise, France. Email address: doukhan@cyu.fr
***Postal address: University of Lyon, UCBL, ISFA LSAF EA2429, 69007 Lyon, France. Email address: yahia.salhi@univ-lyon1.fr
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albrecher, H. and Teugels, J. L. (2007). Asymptotic analysis of a measure of variation. Theory Prob. Math. Statist. 74, 1–10.CrossRefGoogle Scholar
Albrecher, H., Ladoucette, S. A. and Teugels, J. L. (2010). Asymptotics of the sample coefficient of variation and the sample dispersion. J. Statist. Planning Infer. 140, 358–368.CrossRefGoogle Scholar
Bardet, J.-M., Doukhan, P. and Lang, G. (2008). Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle’s estimate. J. Time Ser. Anal. 29, 906–945.CrossRefGoogle Scholar
Bardet, J.-M., Doukhan, P., Lang, G. and Ragache, N. (2008). Dependent Lindeberg central limit theorem and some applications. ESAIM Prob. Statist. 12, 154–172.CrossRefGoogle Scholar
Bernstein, S. (1939). Quelques remarques sur le théorème limite Liapounoff. C. R. (Dokl.) Acad. Sci. USSR 24, 3–8.Google Scholar
Bohk, C., Rau, R. and Cohen, J. E. (2015). Taylor’s power law in human mortality. Demogr. Res. 33, 589–610.CrossRefGoogle Scholar
Brown, M., Cohen, J. and De la Pena, V. (2017). Taylor’s law, via ratios, for some distributions with infinite mean. J. Appl. Prob. 54, 657–669.CrossRefGoogle Scholar
Cohen, J. E. and Saitoh, T. (2016). Population dynamics, synchrony, and environmental quality of Hokkaido voles lead to temporal and spatial Taylor’s laws. Ecology 97, 3402–3413.CrossRefGoogle ScholarPubMed
Cohen, J. E., Davis, R. A. and Samorodnitsky, G. (2020). Heavy-tailed distributions, correlations, kurtosis and Taylor’s law of fluctuation scaling. Proc. R. Soc. A 476, 20200610.CrossRefGoogle Scholar
Cohen, J. E., Plank, M. J. and Law, R. (2012). Taylor’s law and body size in exploited marine ecosystems. Ecol. Evol. 2, 3168–3178.CrossRefGoogle ScholarPubMed
Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications. Stoch. Process. Appl. 106, 63–80.CrossRefGoogle Scholar
Dedecker, J., Doukhan, P., Lang, G., Leon, J. R., Louhichi, S. and Prieur, C. (2007). Weak dependence: models, theory and applications. In Weak Dependence: With Examples and Applications (Lecture Notes Statist. 190), pp. 9–20. Springer, New York.Google Scholar
Doukhan, P. (1994). Mixing: Properties and Examples (Lecture Notes Statist. 85). Springer, New York.Google Scholar
Doukhan, P. (2018). Stochastic Models for Time Series (Mathématiques et Applications 80). Springer.CrossRefGoogle Scholar
Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl. 84, 313–342.CrossRefGoogle Scholar
Doukhan, P. and Neumann, M. H. (2020). Multivariate count autoregression. J. Appl. Prob. 56, 91–115.CrossRefGoogle Scholar
Doukhan, P. and Wintenberger, O. (2008). Weakly dependent chains with infinite memory. Stoch. Process. Appl. 118, 1997–2013.CrossRefGoogle Scholar
Doukhan, P., Fokianos, K. and Rynkiewicz, J. (2021). Mixtures of nonlinear poisson autoregressions. J. Time Ser. Anal. 42, 107–135.CrossRefGoogle Scholar
Doukhan, P., Jakubowicz, J. and Leon, J. R. (2010). Variance estimation with applications. In Dependence in Probability, Analysis and Number Theory: A Volume in Memory of Walter Philipp, eds Berkes, I. et al., pp. 203–232. Kendrick Press.Google Scholar
Doukhan, P., Massart, P. and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Prob. Statist. 30, 63–82.Google Scholar
Eisler, Z., Bartos, I. and Kertész, J. (2008). Fluctuation scaling in complex systems: Taylor’s law and beyond. Adv. Phys. 57, 89–142.CrossRefGoogle Scholar
Fokianos, K., Stove, B., Tjostheim, D. and Doukhan, P. (2020). Multivariate count autoregression. Bernoulli 26, 471–499.CrossRefGoogle Scholar
Kendal, W. and JØrgensen, B. (2011). Tweedie convergence: a mathematical basis for Taylor’s power law, $1/f$ noise, and multifractality. Phys. Rev. E 84, 066120.CrossRefGoogle ScholarPubMed
Leonov, V. P. and Shiryaev, A. N. (1959). On a method of calculation of semi-invariants. Theory Prob. Appl. 4, 319–329.CrossRefGoogle Scholar
Reuman, D. C., Zhao, L., Sheppard, L. W., Reid, P. C. and Cohen, J. E. (2017). Synchrony affects Taylor’s law in theory and data. Proc. Nat. Acad. Sci. USA 114, 6788–6793.CrossRefGoogle Scholar
Rio, E. (2017). Asymptotic Theory of Weakly Dependent Random Processes (Probability Theory and Stochastic Modelling 80). Springer.CrossRefGoogle Scholar
Rosenblatt, M. (1985). Stationary Processes and Random Fields. Birkhäuser, Boston.CrossRefGoogle Scholar
Rosenblatt, M. (2000). Gaussian and Non-Gaussian Linear Time Series and Random Fields (Springer Series in Statistics). Springer, New York.CrossRefGoogle Scholar
Saitoh, T. (2020). Effects of environmental synchrony and density-dependent dispersal on temporal and spatial slopes of Taylor’s law. Popul. Ecol. 62, 300–316.CrossRefGoogle Scholar