On the estimation of the convergence rate in the multidimentional limit theorem for the sum of weakly dependent random variables functions
F.G. Gabbasova, V.T. Dubrovinb, M.S. Fadeevab
aKazan State University of Architecture and Engineering, Kazan, 420043 Russia
bKazan Federal University, Kazan, 420008 Russia
For citation: Gabbasov F.G., Dubrovin V.T., Fadeeva M.S. On the estimation of the convergence rate in the multidimentional limit theorem for the sum of weakly dependent random variables functions. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 266–274.
Для цитирования: Gabbasov F.G., Dubrovin V.T., Fadeeva M.S. On the estimation of the convergence rate in the multidimentional limit theorem for the sum of weakly dependent random variables functions // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 2. – С. 266–274.
Abstract
A refinement of estimates of the convergence rate obtained earlier in the multidimensional central limit theorem for the sums of vectors generated by the sequences of random variables with mixing is close to optimal. This has been achieved by imposing an additional condition on the characteristic functions of these sums, more accurate estimates of the semi-invariants, and using asymptotic expansions for the characteristic functions of the sums of independent random vectors. The result has been obtained using the summation methods for weakly dependent random variables based on S.N. Bernstein's idea of partition of the sums of weakly dependent random variables into long and short partial sums, as a result of which the long sums are almost independent, and the contribution of short sums to the total distribution is small. To estimate the differences between the sum distributions, we have used the S.M. Sadikova's inequality connecting the difference between the characteristic functions of random vectors with the difference between the corresponding distributions. To estimate the contribution of short sums, Markov and Bernstein's inequalities have been used.
Keywords: limit theorem, strong mixing, semi-invariants, asymptotic expansion, convergence rate
Acknowledgements. The work is partially supported by the Russian Foundation for Basic Research (project no. 17-41-160-277).
References
1. Monin A.S., Yaglom M. Statistical Fluid Mechanics: Mechanics of Turbulence. Mineola, N. Y., Dover Publ., 2007. 769 p.
2. Badriev I.B., Zadvornov O.A., Ismagilov L.N., Skvortsov E.V. Solution of plane seepage problems for a multivalued seepage law when there is a point source. J. Appl. Math. Mech., 2009, vol. 73, no. 4, pp. 434–442. doi: 10.1016/j.jappmathmech.2009.08.007.
3. Badriev I.B., Zadvornov O.A. Analysis of the stationary filtration problem with a multivalued law in the presence of a point source. Differ. Equations, 2005, vol. 41, no. 7, pp. 915–922. doi: 10.1007/s10625-005-0231-1.
4. Badriev I.B., Garipova G.Z., Makarov M.V., Paymushin V.N. Numerical solution of the issue about geometrically nonlinear behavior of sandwich plate with transversal soft filler. Res. J. Appl. Sci., 2015, vol. 10, no. 8, pp. 428–435. doi: 10.3923/rjasci.2015.428.435.
5. Badriev I.B., Makarov M.V., Paimushin V.N. Contact statement of mechanical problems of reinforced on a contour sandwich plates with transversally-soft core. Russ. Math., 2017, vol. 61, no. 1, pp. 69–75. doi: 10.3103/S1066369X1701008X.
6. Badriev I.B., Chebakova V.Y., Zheltukhin V.S. Capacitive coupled RF discharge: Modelling at the local statement of the problem. J. Phys.: Conf. Ser., 2017, no. 789 (1), art. 012004, pp. 1–4. doi: 10.1088/1742-6596/789/1/012004.
7. Chebakova V.Ju. Simulation of radio-frequency capacitive discharge at atmospheric pressure in argon. Lobachevskii J. Math., 2017, vol. 38, no. 6, pp. 1165–1178. doi: 10.1134/S1995080217060154.
8. Zheltukhin V.S., Fadeeva M.S., Chebakova V.Ju. Modification of the Scharfetter-Gummel method for calculating the flux of charged particles for simulation of a radio-frequency capacitive coupled discharge. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 444–457. (In Russian)
9. Dubrovin V.T. Convergence rate in limit theorems for weakly dependent random values. Lobachevskii J. Math., 2014, vol. 35, no. 4, pp. 390–396. doi: 10.1134/S1995080214040039.
10. Dubrovin V.T. Central limit theorem for endomorphisms of the Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2006, vol. 148, no. 2, pp. 54–64. (In Russian)
11. Dubrovin V.T. Large deviations in the central limit theorem for endomorphisms of Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2011, vol. 153, no. 1, pp. 195–210. (In Russian)
12. Dubrovin V.T., Gabbasov F.G., Chebakova V.Ju. Multidimensional central limit theorem for sums of functions of the trajectories of endomorphisms. Lobachevskii J. Math., 2016, vol. 37, no. 4, pp. 409–417. doi: 10.1134/S1995080216040053.
13. Gabbasov F.G., Dubrovin V.T., Kugurakov V.S. On the multidimensional limit theorem for endomorphisms of the Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, vol. 157, no. 1, pp. 25–34. (In Russian)
14. Gabbasov F.G., Dubrovin V.T. Estimation of the rate of convergence in the multidimensional central limit theorem for endomorphisms of Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2013, vol. 155, no. 2, pp. 33–43. (In Russian)
15. Gabbasov F.G., Dubrovin V.T. Multi-dimensional limit theorem on large deviations for endomorphisms of Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2014, vol. 156, no. 2, pp. 16–24. (In Russian)
16. Dubrovin V.T., Moskvin D.A. Estimation of senior semi-invariants of sums of weakly dependent quantities. Sov. Math., 1979, vol. 23, no. 5, pp. 18–26.
17. Dubrovin V.T. The central limit theorem for sums of functions of sums of weakly dependent variables. In: Veroyatnostnye metody i kibernetika [Probability Methods and Cybernetics]. Kazan, Izd. Kazan. Univ., 1971, vol. 9, pp. 21–23. (In Russian)
18. Gabbasov F.G. A multidimensional limit theorem for sums of functions of sequences with mixing. Lith. Math. J., 1977, vol. 17, no. 4, pp. 494–505. doi: 10.1007/BF00972271.
19. Sadikova S.M. The distance between distributions associated with their values at the convext sets. Dokl. Akad. Nauk SSSR, 1967, vol. 176, no. 4, pp. 787–789. (In Russian)
20. Sadikova S.M. On the multidimensional central limit theorem. Theory Probab. Its Appl., 1968, vol. 13, no. 1, pp. 164–170. doi: 10.1137/1113015.
Received
October 4, 2017
Gabbasov Farid Gayazovich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Applied Mathematics
Kazan State University of Architecture and Engineering
ul. Zelenaya, 1, Kazan, 420043 Russia
E-mail: gabbasov@kgasu.ru
Dubrovin Vyacheslav Timofeevich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical Statistics
Kazan Federal University
ul. Kremlevskaya, 18, Kazan, 420008 Russia
E-mail: Vyacheslav.Dubrovin@ksu.ru
Fadeeva Maria Sergeevna, Student of the Institute of Computational Mathematics and Information Technologies
Kazan Federal University
ul. Kremlevskaya, 18, Kazan, 420008 Russia
E-mail: manysha-98@mail.ru
Контент доступен под лицензией Creative Commons Attribution 4.0 License.
