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 Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki: Year: Volume: Issue: Page: Find

 Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, Volume 160, Book 2, Pages 266–274 (Mi uzku1451)

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

a Kazan State University of Architecture and Engineering, Kazan, 420043 Russia
b Kazan Federal University, Kazan, 420008 Russia

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-41-160277_ð_à The work is partially supported by the Russian Foundation for Basic Research (project no. 17-41-160-277).

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UDC: 519.21
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Citation: F. G. Gabbasov, V. T. Dubrovin, M. S. Fadeeva, “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, 160, no. 2, Kazan University, Kazan, 2018, 266–274

Citation in format AMSBIB
\Bibitem{GabDubFad18} \by F.~G.~Gabbasov, V.~T.~Dubrovin, M.~S.~Fadeeva \paper On the estimation of the convergence rate in the multidimentional limit theorem for the sum of weakly dependent random variables functions \serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki \yr 2018 \vol 160 \issue 2 \pages 266--274 \publ Kazan University \publaddr Kazan \mathnet{http://mi.mathnet.ru/uzku1451} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000460032400007}