Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T11:32:27.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for a class of tempered subordinators and their inverse processes

Part of: Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  08 January 2021

Nikolai Leonenko ,
Claudio Macci  and
Barbara Pacchiarotti
Nikolai Leonenko
Affiliation:
Cardiff School of Mathematics, Cardiff University, Senghennydd Road Cardiff, CF24 4AG, UK (leonenkon@cardiff.ac.uk)
Claudio Macci
Affiliation:
Cardiff School of Mathematics, Cardiff University, Senghennydd Road Cardiff, CF24 4AG, UK (leonenkon@cardiff.ac.uk)
Barbara Pacchiarotti
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica I-00133 Rome, Italy (macci@mat.uniroma2.it; pacchiar@mat.uniroma2.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper is a non-central moderate deviations result. More precisely we mean a class of large deviation principles that fill the gap between the (trivial) weak convergence of some non-Gaussian identically distributed random variables to their common law, and the convergence of some other related random variables to a constant. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes.

Keywords

Mittag–Leffler functionnon-central moderate deviationsrandom time-changesTweedie distribution

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G52: Stable processes 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 2030 - 2050
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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

Asmussen, S. and Albrecher, H.. Ruin probabilities, 2nd edn (Singapore: World Scientific Publishing, 2010).CrossRefGoogle Scholar
Beghin, L.. On fractional tempered stable processes and their governing differential equations. J. Comput. Phys. 293 (2015), 29–39.CrossRefGoogle Scholar
Berabesi, L., Cerasa, A., Cerioli, A. and Perrotta, D.. A new family of tempered distributions. Electron. J. Stat. 10 (2016), 3871–3893.Google Scholar
Bertoin, J.. Lévy processes (Cambridge: Cambridge University Press, 1996).Google Scholar
Bingham, N. H.. Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 17 (1971), 1–22.CrossRefGoogle Scholar
Capitanelli, R. and D'Ovidio, M.. Delayed and rushed motions through time change. ALEA Lat. Am. J. Probab. Math. Stat. 17 (2020), 183–204.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O.. Large deviations techniques and applications, 2nd edn (New York: Springer, 1998).CrossRefGoogle Scholar
Duffield, N. G. and Whitt, W.. Large deviations of inverse processes with nonlinear scalings. Ann. Appl. Probab. 8 (1998), 995–1026.10.1214/aoap/1028903372CrossRefGoogle Scholar
Gajda, J. and Wyłomańska, A.. Geometric Brownian motion with tempered stable waiting times. J. Stat. Phys. 148 (2012), 296–305.CrossRefGoogle Scholar
Gajda, J., Kumar, A. and Wyłomaśka, A.. Stable Lévy process delayed by tempered stable subordinator. Stat. Probab. Lett. 145 (2019), 284–292.CrossRefGoogle Scholar
Glynn, P. and Whitt, W.. Large deviations behavior of counting processes and their inverses. Queueing Syst. Theory Appl. 17 (1994), 107–128.CrossRefGoogle Scholar
Grabchak, M.. Tempered stable distributions: stochastic models for multiscale processes (Cham: Springer, 2016).CrossRefGoogle Scholar
Gupta, N., Kumar, A. and Leonenko, N.. Mixtures of tempered stable subordinators, arXiv:1905.00192, Math. Commun., to appear (2021+).Google Scholar
Gulinsky, O. V. and Veretennikov, A. Y.. Large deviations for discrete-time processes with averaging (Utrecht: VSP, 1993).10.1515/9783110917802CrossRefGoogle Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J.. Theory and applications of fractional differential equations (Amsterdam: Elsevier, 2006).Google Scholar
Klebanov, L. and Slamova, L.. Tempered distributions: does universal tempering procedure exist?, arXiv:1505.02068, 2015.Google Scholar
Küchler, U. and Tappe, S.. Tempered stable distributions and processes. Stoch. Process. Appl. 123 (2013), 4256–4293.CrossRefGoogle Scholar
Kumar, A. and Vellaisamy, P.. Inverse tempered stable subordinators. Stat. Probab. Lett. 103 (2015), 134–141.CrossRefGoogle Scholar
Kumar, A., Gajda, J., Wyłomańska, A. and Połoczaśki, R.. Fractional Brownian motion delayed by tempered and inverse tempered stable subordinators. Methodol. Comput. Appl. Probab. 21 (2019), 185–202.CrossRefGoogle Scholar
Kumar, A., Upadhye, N. S., Wyłomańska, A. and Gajda, J.. Tempered Mittag–Leffler Lévy processes. Commun. Stat. Theory Methods 48 (2019), 396–411.CrossRefGoogle Scholar
Kumar, A., Leonenko, N. and Pichler, A.. Fractional risk process in insurance. Math. Financ. Econ. 14 (2020), 43–65.CrossRefGoogle Scholar
Mainardi, F., Mura, A. and Pagnini, G.. The functions of the Wright type in fractional calculus, Lecture Notes of Seminario Interdisciplinare di Matematica, Vol. 9, Universita degli Studi della Basilicata, pp. 111–128 (2010)Google Scholar
Rosinski, J.. Tempering stable processes. Stoch. Process. Appl. 117 (2007), 677–707.CrossRefGoogle Scholar
Samorodnitsky, G.. Stochastic processes and long range dependence (Cham: Springer, 2016).CrossRefGoogle Scholar
Sato, K.. Lévy processes and infinitely divisible distributions (Cambridge: Cambridge University Press, 1999).Google Scholar
Veretennikov, A. Y.. On large deviations for additive functionals of Markov processes. I. Teor. Veroyatnost. i Primenen. 38 (1993), 758–774; English translation in Theory Probab. Appl. 38 (1993), 706–719.Google Scholar
Wyłomańska, A.. Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes. Physica A 391 (2012), 5685–5696.CrossRefGoogle Scholar
Wyłomańska, A.. The tempered stable process with infinitely divisible inverse subordinators. J. Stat. Mech. Theory Exp. 2013 (2013), Article P10011, 18 pp.CrossRefGoogle Scholar