Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T17:34:50.180Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Distribution theory - Probability Nonparametric inference Limit theorems

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
Get access Rights & Permissions [Opens in a new window]

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 

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

[1] Arcones, M. A. (2002).Large and moderate deviations of empirical processes with nonstandard rates.Statist. Prob. Lett. 57,315–326.Google Scholar
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation.Cambridge University Press.Google Scholar
[3] Borovkov, A. A. and Borovkov, K. A. (2008).Asymptotic Analysis of Random Walks.Cambridge University Press.Google Scholar
[4] Borovkov, A. A. and Mogul′skiĭ, A. A. (2006).Integro-local and integral theorems for sums of random variables with semiexponential distributions. Sibisrk. Mat. Zh. 47, 1218–1257 (in Russian). English translation: Sib. Math. J. 47,990–1026.Google Scholar
[5] Daley, D. J. and Goldie, C. M. (2006).The moment index of minima. II..Statist. Prob. Lett. 76,831–837.Google Scholar
[6] Dembo, A. and Zeitouni, O. (2009).Large Deviations Techniques and Applications, 2nd edn.Springer,Berlin.Google Scholar
[7] Denisov, D. É (2006).On the existence of a regularly varying majorant of an integrable monotone function. Mat. Zametki 79 142–145 (in Russian). English translation: Math. Notes 79,129–133.Google Scholar
[8] Duffy, K. R. and Sapozhnikov, A. (2008).The large deviation principle for the on-off Weibull sojourn process.J. Appl. Prob. 45,107–117.Google Scholar
[9] Foss, S., Korshunov, D. and Zachary, S. (2013).An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn.Springer,New York.Google Scholar
[10] Galambos, J. and Seneta, E. (1973).Regularly varying sequences.Proc. Amer. Math. Soc. 41,110–116.Google Scholar
[11] Gantert, N. (1998).Functional Erdős–Renyi laws for semiexponential random variables.Ann. Prob. 26,1356–1369.Google Scholar
[12] Gantert, N. (2000).A note on logarithmic tail asymptotics and mixing.Statist. Prob. Lett. 49,113–118.Google Scholar
[13] Gantert, N., Kavita, R. and Franz, R. (2014).Large deviations for weighted sums of stretched exponential random variables.Electron. Commun. Prob. 19, 14pp.Google Scholar
[14] Iksanov, A. M. and Rösler, U. (2006).Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities.Ukraïn. Mat. Zh. 58, 451–471 (in Ukranian). English translation: Ukrainian Math. J. 58,505–528.Google Scholar
[15] Lehtomaa, J. (2015).Limiting behaviour of constrained sums of two variables and the principle of a single big jump.Statist. Prob. Lett. 107,157–163.Google Scholar
[16] Mikosch, T. and Nagaev, A. V. (1998).Large deviations of heavy-tailed sums with applications in insurance.Extremes 1,81–110.Google Scholar
[17] Nagaev, S. V. (1979).Large deviations of sums of independent random variables.Ann. Prob. 7,745–789.Google Scholar
[18] Stabile, G. and Torrisi, G. L. (2010).Large deviations of Poisson shot noise processes under heavy tail semi-exponential conditions.Statist. Prob. Lett. 80,1200–1209.Google Scholar
[19] Varadhan, S. R. S. (2008).Large deviations.Ann. Prob. 36,397–419.CrossRefGoogle Scholar