On tails of symmetric and totally asymmetric alpha-stable distributions

• Autorzy: Bednorz Witold M. , Łochowski Rafał M. , Martynek Rafał

Identyfikatory

DOI

10.37190/0208-4147.41.2.7

• Języki publikacji: EN

Abstrakty

EN

We estimate up to universal constants tails of symmetric and to-tally asymmetric 1-dimensional α-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of α close to 2we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution.

Słowa kluczowe

EN

alpha-stable distributions; tail estimates

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2021
• Tom: Vol. 41, Fasc. 2
• Strony: 321--345
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Bednorz Witold M.

• University of Warsaw Banacha 2 Warszawa, Poland

autor

Łochowski Rafał M.

• Warsaw School of Economics Al. Niepodległoś́ci 162, Warszawa, Poland

autor

Martynek Rafał

• University of Warsaw Banacha 2 Warszawa, Poland

Bibliografia

• [1] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math.Soc. 95 (1960), 263-273.
• [2] K. Bogdan, T. Grzywny and M. Ryznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (2014), 3543-3571.
• [3] K. Bogdan, T. Grzywny and M. Ryznar, Density and tails of unimodal convolution semigroups, arXiv:1305.0976v1 (2013).
• [4] T. Grzywny and K. Szczypkowski, Estimates of heat kernels of non-symmetric Lévy processes, arXiv:1710.07793v2 (2020).
• [5] O. Kallengerg, Foundations of Modern Probability, Springer, New York, 2002.
• [6] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, 2nd ed., Wiley, New York, 1971.
• [7] J. P. Nolan, Numerical calculation of stable densities and distribution functions, Comm. Statist. Stoch. Models 13 (1997), 759-774.
• [8] G. Pólya, On the zeros of an integral function represented by Fourier’s integral, Messenger Math. 52 (1923), 185-188.
• [9] W. E. Pruitt, The growth of random walks and Lévy processes, Ann. Probab. 9 (1981), 948-956.
• [10] J. Rosiński, On the series representation of infinitely divisible random vectors, Ann. Probab.18 (1990), 405-430.
• [11] J. Rosiński, Simulations of Lévy processes, in: Encyclopedia of Statistics in Quality and Reliability: Computationally Intensive Methods and Simulation, Wiley, 2008.
• [12] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, 1994.
• [13] P. Sztonyk, Transition density estimates for jump Lévy processes, Stochastic Process. Appl. 121(2011), 1245-1265.
• [14] M. Talagrand, Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems, Ergeb. Math. Grenzgeb. 60, Springer, New York, 2014.
• [15] V. Uchaikin and V. Zolotarev, Chance and Stability: Stable Distributions and their Applications, De Gruyter, Berlin, 2011.
• [16] T. Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Trans. Amer. Math. Soc. 359 (2007), 2851-2879.
• [17] V. M. Zolotarev, One-Dimensional Stable Distributions, Transl. Math. Monogr. 65, Amer. Math. Soc., Providence, RI, 1986.