Abstract:
In this paper we modify the Stein method and the auxiliary technique of
distributional transformations of random variables. This enables us to
estimate the convergence rate of distributions of normalized geometric sums
to the Laplace law. For independent summands, the developed approach provides
an optimal estimate involving the ideal metric of order 3. New results are
also obtained for the Kolmogorov and Kantorovich metrics.
Keywords:
Stein's method, geometric random sum, zero-bias transform, equilibrium transform,
convergence rate to the Laplace distribution, analogue of the Berry–Esseen inequality, optimal estimate.
Citation:
N. A. Slepov, “Convergence rate of random geometric sum distributions to the Laplace law”, Teor. Veroyatnost. i Primenen., 66:1 (2021), 149–174; Theory Probab. Appl., 66:1 (2021), 121–141
\Bibitem{Sle21}
\by N.~A.~Slepov
\paper Convergence rate of random geometric sum distributions to the Laplace law
\jour Teor. Veroyatnost. i Primenen.
\yr 2021
\vol 66
\issue 1
\pages 149--174
\mathnet{http://mi.mathnet.ru/tvp5363}
\crossref{https://doi.org/10.4213/tvp5363}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4213093}
\zmath{https://zbmath.org/?q=an:1466.62258}
\transl
\jour Theory Probab. Appl.
\yr 2021
\vol 66
\issue 1
\pages 121--141
\crossref{https://doi.org/10.1137/S0040585X97T990290}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85108799787}
Linking options:
https://www.mathnet.ru/eng/tvp5363
https://doi.org/10.4213/tvp5363
https://www.mathnet.ru/eng/tvp/v66/i1/p149
This publication is cited in the following 5 articles:
K. Barman, N. S. Upadhye, “On Stein factors for Laplace approximation and their application to random sums”, Statistics & Probability Letters, 206 (2024), 109996
M. S. Tikhov, “Otsenivanie raspredelenii po vyborkam sluchainogo ob'ema”, Vestnik TvGU. Seriya: Prikladnaya matematika, 2023, no. 4, 5–24
A. Kudryavtsev, O. Shestakov, “Estimates of the convergence rate in the generalized Rényi theorem with a structural digamma distribution using zeta metrics”, Mathematics, 11:21 (2023), 4477
A. Bulinski, N. Slepov, “Sharp estimates for proximity of geometric and related sums distributions to limit laws”, Mathematics, 10:24 (2022), 4747
Q. Liu, A. Xia, “Geometric sums, size biasing and zero biasing”, Electron. Commun. Probab., 27 (2022), 1-113
Citation in format AMSBIB
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