Estimation of the rate of convergence in the central limit theorem for a sequence of series in terms of averaged pseudomoments
Authors:
M. M. Kapustei and P. V. Slyusarchuk
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 99 (2019), 101-111
MSC (2010):
Primary 60F05
DOI:
https://doi.org/10.1090/tpms/1083
Published electronically:
February 27, 2020
MathSciNet review:
3908659
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Additional Information
Abstract: A generalization of Zolotarev’s estimates is proved for a sequence of series of random variables. The estimates are expressed in terms of averaged pseudomoments.
References
- V. M. Zolotarev, On the closeness of the distributions of two sums of independent random variables, Teor. Verojatnost. i Primenen. 10 (1965), 519–526 (Russian, with English summary). MR 0189109
- V. Paulauskas, A certain strengthening of Ljapunov’s theorem, Litovsk. Mat. Sb. 9 (1969), 323–328 (Russian, with Lithuanian and English summaries). MR 0266280
- V. M. Zolotarev, Exactness of an approximation in the central limit theorem, Proceedings of the Second Japan-USSR Symposium on Probability Theory (Kyoto, 1972) Springer, Berlin, 1973, pp. 531–543. Lecture Notes in Math., Vol. 330. MR 0443048
- V. M. Zolotarev, Sovremennaya teoriya summirovaniya nezavisimykh sluchaĭ nykh velichin, Teoriya Veroyatnosteĭ i Matematicheskaya Statistika. [Probability Theory and Mathematical Statistics], “Nauka”, Moscow, 1986 (Russian). MR 917274
- Yuliya Mishura, Yevheniya Munchak, and Petro Slyusarchuk, The rate of convergence to the normal law in terms of pseudomoments, Mod. Stoch. Theory Appl. 2 (2015), no. 2, 95–106. MR 3389584, DOI https://doi.org/10.15559/15-vmsta23
- Yu. S. Mīshura and Ē. Yu. Munchak, Rate of convergence of option prices using the method of pseudomoments, Teor. Ĭmovīr. Mat. Stat. 92 (2015), 110–124 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 92 (2016), 117–133. MR 3553430, DOI https://doi.org/10.1090/tpms/987
- P. V. Slyusarchuk and Ī. Ĭ. Polyak, A generalization of a result of V. M. Zolotarev, Nauk. Vīsn. Uzhgorod. Univ. Ser. Mat. 3 (1998), 184–189 (Ukrainian, with English and Ukrainian summaries). MR 2498517
- T. V. Boyarishcheva and P. V. Slyusarchuk, An estimate for the rate of convergence in the central limit theorem for nonidentically distributed random variables, Nauk. Vīsn. Uzhgorod. Univ. Ser. Mat. 4 (1999), 12–16 (Ukrainian, with English and Ukrainian summaries). MR 2498576
- Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
References
- V. M. Zolotarev, On the closeness of the distribution of two sums of independent random variables, Teor. Veroyatn. Primenen. 10 (1965), 519–526; English transl. in Theory Probab. Appl. 10 (1965), 472–479. MR 0189109
- V. Paulauskas, On a strengthening of a Lyapunov theorem, Litovsk. Matem. Zh. 9 (1969), no. 2, 323–328. (Russian) MR 0266280
- V. M. Zolotarev, Exactness of an approximation in the central limit theorem, Proceedings of the Second Japan–USSR Symposium on Probability Theory, Lecture Notes in Mathematics, vol. 330, Springer-Verlag, Berlin, 1973, pp. 531–543. MR 0443048
- V. M. Zolotarev, Modern Theory of Summation of Independent Random Variables, “Nauka”, Moscow, 1986. (Russian) MR 917274
- Yu. Mishura, Ye. Munchak, and P. Slyusarchuk, The rate of convergence to the normal law in terms of pseudomoments, Modern Stochastics: Theory and Applications 2 (2015), no. 2, 95–106. MR 3389584
- Yu. Mishura and Ye. Munchak, The convergence rate of stock prices using the method of pseudo-moments, Teor. Imovir. Matem. Statyst. 92 (2015), 110–124; English transl. in Theor. Probab. Math. Statist. 92 (2016), 117–133. MR 3553430
- P. V. Slyusarchuk and I. Y. Polyak, A generalization of a result of V. M. Zolotarev, Nauk. Vistn. Uzhorod. Univer. Ser. Matem. 3 (1998), 184–189. (Ukrainian) MR 2498517
- T. V. Boyarishcheva and P. V. Slyusarchuk, An estimate of the rate of convergence in the central limit theorem for non-identically distributed variables, Nauk. Vistn. Uzhorod. Univer. Ser. Matem. 4 (1999), 12–16. (Ukrainian) MR 2498576
- M. Loève, Probability Theory, D. Van Nostrand, Princeton, New Jersey, 1955. MR 0203748
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Additional Information
M. M. Kapustei
Affiliation:
Department of Probability Theory and Mathematical Analysis, Faculty for Mathematics, Uzhgorod National University, Universytets’ka Street, 14, Uzhgorod, 88020 Ukraine
Email:
michaelkapustey@gmail.com
P. V. Slyusarchuk
Affiliation:
Department of Probability Theory and Mathematical Analysis, Faculty for Mathematics, Uzhgorod National University, Universytets’ka Street, 14, Uzhgorod, 88020 Ukraine
Email:
petro_slyusarchuk@ukr.net
Keywords:
Convergence,
central limit theorem,
pseudomoments,
sequence of series,
nonidentically distributed random variables
Received by editor(s):
June 25, 2018
Published electronically:
February 27, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
