Noncentral moderate deviations for time-changed Lévy processes with inverse of stable subordinators
Volume 12, Issue 2 (2025), pp. 203–224
Pub. online: 31 December 2024
Type: Research Article
Open Access
Open Access
Received
7 August 2024
7 August 2024
Revised
16 December 2024
16 December 2024
Accepted
17 December 2024
17 December 2024
Published
31 December 2024
31 December 2024
Abstract
This paper presents some extensions of recent noncentral moderate deviation results. In the first part, the results in [Statist. Probab. Lett. 185, Paper No. 109424, 8 pp. (2022)] are generalized by considering a general Lévy process $\{S(t):t\ge 0\}$ instead of a compound Poisson process. In the second part, it is assumed that $\{S(t):t\ge 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\ge 0\}$ can be seen as the difference of two independent nonnull subordinators. In this way, the results in [Mod. Stoch. Theory Appl. 11, 43–61] for Skellam processes are generalized.
References
Beghin, L., Macci, C.: Non-central moderate deviations for compound fractional Poisson processes. Statist. Probab. Lett. 185 (2022). Paper No. 109424, 8 pp.. MR4383208. https://doi.org/10.1016/j.spl.2022.109424
Chi, Z.: On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation. Stochastic Processes Appl. 126, 1124–1144 (2016). MR3461193. https://doi.org/10.1016/j.spa.2015.11.001
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998). MR1619036. https://doi.org/10.1007/978-1-4612-5320-4
Giuliano, R., Macci, C.: Some examples of noncentral moderate deviations for sequences of real random variables. Mod. Stoch. Theory Appl. 10, 111–144 (2023). MR4573675. https://doi.org/10.15559/23-vmsta219
Glynn, P.W., Whitt, W.: Large deviations behavior of counting processes and their inverses. Queueing Syst. Theory Appl. 17, 107–128 (1994). MR1295409. https://doi.org/10.1007/BF01158691
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, New York (2014). MR3244285. https://doi.org/10.1007/978-3-662-43930-2
Iafrate, F., Macci, C.: Asymptotic results for the last zero crossing time of a Brownian motion with non-null drift. Stat. Probab. Lett. 166, Paper no. 108881, 9 pp. (2020). MR4129103. https://doi.org/10.1016/j.spl.2020.108881
Kerss, A., Leonenko, N.N., Sikorskii, A.: Fractional Skellam processes with applications to finance. Fract. Calc. Appl. Anal. 17, 532–551 (2014). MR3181070. https://doi.org/10.2478/s13540-014-0184-2
Lee, J., Macci, C.: Noncentral moderate deviations for fractional Skellam processes. Mod. Stoch. Theory Appl. 11, 43–61 (2024). MR4692017. https://doi.org/10.15559/23-VMSTA235
Leonenko, N., Macci, C., Pacchiarotti, B.: Large deviations for a class of tempered subordinators and their inverse processes. Proc. R. Soc. Edinb., Sect. A 151, 2030–2050 (2021). MR4349433. https://doi.org/10.1017/prm.2020.95
Meerschaert, M.M., Nane, E., Vellaisamy, P.: The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. 16, 1600–1620 (2011). MR2835248. https://doi.org/10.1214/EJP.v16-920
Meerschaert, M.M., Sikorskii, A.: Stochastic models for fractional calculus, Walter de Gruyter & Co., Berlin (2012). MR2884383. https://doi.org/10.1515/978-3-110-25816-5
