Abstract:
New bounds (analogous to the bounds obtained by Kolmogorov, Rogozin and Esseen) are derived for the concentration function of the sums of independent random variables with values in a Hilbert space. In particular, the absolute constants used in the estimates don't depend on the dimension in the finite-dimensional space. Further, some limit theorems for the concentration function and some estimates for the concentration functions
of infinitely divisible distributions are given.
Citation:
G. Siegel, “Upper bounds for the concentration function in a Hilbert space”, Teor. Veroyatnost. i Primenen., 26:2 (1981), 335–349; Theory Probab. Appl., 26:2 (1982), 328–343
\Bibitem{Sie81}
\by G.~Siegel
\paper Upper bounds for the concentration function in a~Hilbert space
\jour Teor. Veroyatnost. i Primenen.
\yr 1981
\vol 26
\issue 2
\pages 335--349
\mathnet{http://mi.mathnet.ru/tvp2514}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=616624}
\zmath{https://zbmath.org/?q=an:0501.60021|0487.60014}
\transl
\jour Theory Probab. Appl.
\yr 1982
\vol 26
\issue 2
\pages 328--343
\crossref{https://doi.org/10.1137/1126032}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1981NJ71600007}
Linking options:
https://www.mathnet.ru/eng/tvp2514
https://www.mathnet.ru/eng/tvp/v26/i2/p335
This publication is cited in the following 11 articles:
S. M. Ananevskii, “O konstantakh v neravenstvakh Kolmogorova–Rogozina i Kestena v gilbertovom prostranstve”, Veroyatnost i statistika. 30, Zap. nauchn. sem. POMI, 501, POMI, SPb., 2021, 8–23
Chi Zh., “On a Multivariate Strong Renewal Theorem”, J. Theor. Probab., 31:3 (2018), 1235–1272
Ya. S. Golikova, “Ob uluchshenii otsenki rasstoyaniya mezhdu raspredeleniyami posledovatelnykh summ nezavisimykh sluchainykh velichin”, Veroyatnost i statistika. 27, Zap. nauchn. sem. POMI, 474, POMI, SPb., 2018, 118–123
A. L. Miroshnikov, N. V. Miller, N. I. Popova, Yu. V. Shvets, “O nekotorykh voprosakh integrirovaniya v mnogomernykh prostranstvakh”, Mezhdunar. nauch.-issled. zhurn., 2017, no. 12-5(66), 30–35
E. L. Maistrenko, “Estimation of the constant in the inequality for the uniform distance between distributions of sequential sums of i.i.d. random variables”, J. Math. Sci. (N. Y.), 229:6 (2018), 741–743
Yu. S. Eliseeva, “Multivariate estimates for the concentration functions of weighted sums of independent identically distributed random variables”, J. Math. Sci. (N. Y.), 204:1 (2015), 78–89
A. L. Miroshnikov, “Estimates of the Multidimensional Levy's Concentration Function”, Theory Probab. Appl., 34:3 (1989), 535–540
A. P. Suchkov, N. G. Ushakov, “Exponential Bounds for Decreasing of the Maximal Probability in Infinite Dimensional Spaces”, Theory Probab. Appl., 34:4 (1989), 735–738
A. G. Postnikov, A. A. Yudin, “An Estimate of the Maximum Probability for the Sum of Independent Vectors”, Theory Probab. Appl., 32:2 (1987), 331–334
G. Siegel, “Vague convergence for convolutions and for infinitely divisible functions”, Theory Probab. Appl., 31:1 (1987), 152–159
N. G. Ušakov, “Upper bounds for the maximum probability for sums of independent random vectors”, Theory Probab. Appl., 30:1 (1986), 38–49
Citation in format AMSBIB
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