**Author details**

Victor Korolev1,2,3\* and Alexander Zeifman3,4

1 Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

2 Hangzhou Dianzi University, Hangzhou, China

3 Institute of Informatics Problems, Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, Moscow, Russia

**References**

1998

2002

Hall; 1994

**40**(3):803-806

**49**(3):377-391

**187**

[1] Shiryaev AN. Foundations of Financial Mathematics. Vol. 1. Facts, Models. – Singapore: World Scientific;

*DOI: http://dx.doi.org/10.5772/intechopen.89659*

Applications. Boca Raton: CRC Press;

[11] Gnedenko BV. On estimation of the unknown parameters of distributions from a random number of independent observations. Probability Theory and Mathematical Statistics. Proccedings of Tbilisi Mathematical Institute named after A. M. Razmadze. 1989;**24**:146-150

[12] Kolmogorov AN. The method of median in the theory of errors.

Matematicheskii Sbornik. 1931;**38**(3/4):

[13] Korolev VY, Kossova EV. On limit distributions of randomly indexed multidimensional random sequences with an operator normalization. Journal of Mathematical Sciences. 1992;**72**(1):

[14] Korolev VY, Kossova EV. Convergence of multidimensional random sequences with independent

random indices. Journal of

Mathematical Sciences. 1995;**76**(2):

[15] Zolotarev VM. One-Dimensional Stable Distributions. Providence: American Mathematical Society; 1986

distributions. Journal of Mathematical Sciences. 2001;**105**(6):2569-2576

[17] Korolev VY, Zeifman AI. A note on mixture representations for the Linnik and Mittag-Leffler distributions and

Mathematical Sciences. 2017;**218**(3):

Convergence of statistics constructed from samples with random sizes to the Linnik and Mittag-Leffler distributions

[16] Kolokoltsov V, Korolev V, Uchaikin V. Fractional stable

their applications. Journal of

[18] Korolev VY, Zeifman AI.

1996

*From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums…*

(in Russian)

47-50

2915-2929

2259-2268

314-327

[2] Bening V, Korolev V. Generalized Poisson Models and Their Application in Insurance and Finance. Utrecht: VSP;

[3] Meerschaert MM, Scheffler H-P. Limit theorems for continuous-time random walks with infinite mean waiting times. Journal of Applied Probability. 2004;**41**(3):623-638

[4] Samorodnitsky G, Taqqu MS. Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance. New York: Chapman and

[5] Chen J. From the central limit theorem to heavy-tailed distributions. Journal of Applied Probability. 2003;

[7] Schluter C, Trede M. Weak

distributions. Journal of Applied Probability. 2016;**53**:121-129

[8] Bening V, Korolev V. On an application of the Student distribution in the theory of probability and mathematical statistics. Theory of Probability and its Applications. 2005;

problems related to the Laplace distribution. Informatics and its Apllications. 2008;**2**(2):19-34

[6] Korolev VY. Convergence of random sequences with independent random indices. I. Theory of Probability and its Applications. 1994;**39**(2):313-333

convergence to the Student and Laplace

[9] Bening V, Korolev V. Some statistical

[10] Gnedenko BV, Korolev V. Random Summation: Limit Theorems and

4 Vologda State University, Vologda, Russia

\*Address all correspondence to: vkorolev@cs.msu.ru

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums… DOI: http://dx.doi.org/10.5772/intechopen.89659*
