1. Home
  2. To appear
  3. Approximations related to tempered stabl ...

Modern Stochastics: Theory and Applications


Approximations related to tempered stable distributions
Pub. online: 27 February 2025      Type: Research Article      Open accessOpen Access
Received
14 August 2024
Revised
24 January 2025
Accepted
15 February 2025
Published
27 February 2025

Abstract

In this article, we first obtain, for the Kolmogorov distance, an error bound between a tempered stable and a compound Poisson distribution (CPD) and also an error bound between a tempered stable and an α-stable distribution via Stein’s method. For the smooth Wasserstein distance, an error bound between two tempered stable distributions (TSDs) is also derived. As examples, we discuss the approximation of a TSD to normal and variance-gamma distributions (VGDs). As corollaries, the corresponding limit theorem follows.

References

[1] 
Arras, B., Houdré, C.: On Stein’s Method for Infinitely Divisible Laws with Finite First Moment. Springer (2019) MR3931309
[2] 
Barbour, A.D.: Stein’s method for diffusion approximations. Probab. Theory Relat. Fields 84(3), 297–322 (1990) MR1035659. https://doi.org/10.1007/BF01197887
[3] 
Blumenthal, R.M., Getoor, R.K.: Sample functions of stochastic processes with stationary independent increments. J. Math. Mech., 493–516 (1961) MR0123362
[4] 
Boyarchenko, S.I., Levendorskiǐ, S.Z.: Option pricing for truncated lévy processes. Int. J. Theor. Appl. Finance 3(03), 549–552 (2000) MR3951952. https://doi.org/10.1142/S0219024919500110
[5] 
Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: An empirical investigation. J. Bus. 75(2), 305–332 (2002)
[6] 
Čekanavičius, V.: Approximation Methods in Probability Theory. Springer (2016) MR3467748. https://doi.org/10.1007/978-3-319-34072-2
[7] 
Čekanavičius, V., Vellaisamy, P.: Approximating by convolution of the normal and compound poisson laws via stein’s method. Lith. Math. J. 58, 127–140 (2018) MR3814710. https://doi.org/10.1007/s10986-018-9392-5
[8] 
Chen, P., Xu, L.: Approximation to stable law by the lindeberg principle. J. Appl. Math. Anal. Appl. 480(2), 123338 (2019) MR4000076. https://doi.org/10.1016/j.jmaa.2019.07.028
[9] 
Chen, P., Nourdin, I., Xu, L., Yang, X., Zhang, R.: Non-integrable stable approximation by stein’s method. J. Theor. Probab., 1–50 (2022) MR4414414. https://doi.org/10.1007/s10959-021-01094-5
[10] 
Döbler, C., Gaunt, R.E., Vollmer, S.J.: An iterative technique for bounding derivatives of solutions of stein equations (2017) MR3724564. https://doi.org/10.1214/17-EJP118
[11] 
Gaunt, R.E.: On stein’s method for products of normal random variables and zero bias couplings. Bernoulli 23(4B), 3311–3345 (2017) MR3654808. https://doi.org/10.3150/16-BEJ848
[12] 
Gaunt, R.E.: Variance-gamma approximation via stein’s method. Electron. J. Probab. 19(38), 1–33 (2014) MR3194737. https://doi.org/10.1214/EJP.v19-3020
[13] 
Gaunt, R.E.: Wasserstein and kolmogorov error bounds for variance-gamma approximation via stein’s method i. J. Theor. Probab. 33(1), 465–505 (2020) MR4064309. https://doi.org/10.1007/s10959-018-0867-4
[14] 
Gaunt, R.E., Li, S.: Bounding kolmogorov distances through wasserstein and related integral probability metrics. J. Appl. Math. Anal. Appl. 522(1), 126985 (2023) MR4533915. https://doi.org/10.1016/j.jmaa.2022.126985
[15] 
Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables vol. 2420. Addison-wesley (1968) MR0233400
[16] 
Goldstein, L., Reinert, G.: Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7(4), 935–952 (1997) MR1484792. https://doi.org/10.1214/aoap/1043862419
[17] 
Houdré, C., Pérez-Abreu, V., Surgailis, D.: Interpolation, correlation identities, and inequalities for infinitely divisible variables. J. Fourier Anal. Appl. 4, 651–668 (1998) MR1665993. https://doi.org/10.1007/BF02479672
[18] 
Jin, X., Li, X., Lu, J.: A kernel bound for non-symmetric stable distribution and its applications. J. Appl. Math. Anal. Appl. 488(2), 124063 (2020) MR4081550. https://doi.org/10.1016/j.jmaa.2020.124063
[19] 
Ken-Iti, S.: Lévy Processes and Infinitely Divisible Distributions vol. 68. Cambridge university press (1999)
[20] 
Kesavan, S.: Topics in functional analysis and applications. Wiley Eastern Ltd. (1989) MR0990018
[21] 
Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J.: Financial market models with lévy processes and time-varying volatility. J. Bank. Finance 32(7), 1363–1378 (2008)
[22] 
Koponen, I.: Analytic approach to the problem of convergence of truncated lévy flights towards the gaussian stochastic process. Phys. Rev. E 52(1), 1197 (1995)
[23] 
Küchler, U., Tappe, S.: Bilateral gamma distributions and processes in financial mathematics. Stoch. Process. Appl. 118(2), 261–283 (2008). doi:https://doi.org/10.1016/j.spa.2007.04.006. MR2376902
[24] 
Küchler, U., Tappe, S.: Tempered stable distributions and processes. Stoch. Process. Appl. 123(12), 4256–4293 (2013). doi:https://doi.org/10.1016/j.spa.2013.06.012. MR3096354
[25] 
Kumar, A., Vellaisamy, P., Viens, F.: Poisson approximation to the convolution of psds. Probab. Math. Stat. 42, 63–80 (2022) MR4490669. https://doi.org/10.37190/0208-4147.00056
[26] 
Ley, C., Reinert, G., Swan, Y.: Stein’s method for comparison of univariate distributions. Probab. Surv. 14, 1–52 (2017) MR3595350. https://doi.org/10.1214/16-PS278
[27] 
Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus: from Stein’s Method to Universality vol. 192. Cambridge University Press (2012) MR2962301. https://doi.org/10.1017/CBO9781139084659
[28] 
Rachev, S.T., Kim, Y.S., Bianchi, M.L., Fabozzi, F.J.: Financial Models with Lévy Processes and Volatility Clustering. John Wiley & Sons (2011)
[29] 
Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117(6), 677–707 (2007) MR2327834. https://doi.org/10.1016/j.spa.2006.10.003
[30] 
S Upadhye, N., Barman, K.: A unified approach to stein’s method for stable distributions. Probab. Surv. 19, 533–589 (2022) MR4507474. https://doi.org/10.1214/20-ps354
[31] 
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, vol. 6, pp. 583–603 (1972). University of California Press MR0402873
[32] 
Stein, E.M., Shakarchi, R.: Fourier Analysis: an Introduction vol. 1. Princeton University Press (2011) MR1970295
[33] 
Sztonyk, P.: Estimates of tempered stable densities. J. Theor. Probab. 23(1), 127–147 (2010) MR2591907. https://doi.org/10.1007/s10959-009-0208-8
[34] 
Upadhye, N.S., Čekanavičius, V., Vellaisamy, P.: On Stein operators for discrete approximations. Bernoulli 23(4A), 2828–2859 (2017). doi:https://doi.org/10.3150/16-BEJ829. MR3648047
[35] 
Vellaisamy, P., Čekanavičius, V.: Infinitely divisible approximations for sums of m-dependent random variables. J. Theor. Probab. 31, 2432–2445 (2018) MR3866620. https://doi.org/10.1007/s10959-017-0774-0
[36] 
Villani, C.: Topics in Optimal Transportation vol. 58. American Mathematical Soc. (2021) MR1964483. https://doi.org/10.1090/gsm/058
[37] 
Wang, Y., Yuan, C.: Convergence of the euler-maruyama method for stochastic differential equations with respect to semimartingales. Appl. Math. Sci. (Ruse) 1(41-44), 2063–2077 (2007) MR2369925
[38] 
Xu, L.: Approximation of stable law in wasserstein-1 distance by stein’s method. Ann. Appl. Probab. 29(1), 458–504 (2019) MR3910009. https://doi.org/10.1214/18-AAP1424

Full article Related articles PDF XML
Full article Related articles PDF XML

Copyright
© 2025 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Probability approximations tempered stable distributions stable distributions Stein’s method characteristic function approach

MSC2020
62E17 (primary) 62E20 (primary) 60E05 (secondary) 60E07 (secondary)

Metrics
since March 2018
295

Article info
views

103

Full article
views

87

PDF
downloads

23

XML
downloads


Share


RSS