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Bounds for the accuracy of Poissonian approximations of stable laws

Author

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  • Bentkus, V.
  • Götze, F.
  • Paulauskas, V.

Abstract

Stable law Gz admit a well-known series representation of the type where [Gamma]1, [Gamma]2, ... are the successive times of jumps of a standard Poisson process, and X1, X2, ..., denote i.i.d. random variables, independent of [Gamma]1, [Gamma]2, ... We investigate the rate of approximation of G[alpha] by distributions of partial sums Sn = [summation operator]nj = 1 [Gamma]-1/[alpha]jXj, and we get (asymptotically) optimal bounds for the variation of . The results obtained complement and improve the results of A. Janicki and P. Kokoszka, and M. Ledoux and V. Paulauskas. Bounds for the concentration function of Sn are also proved.

Suggested Citation

  • Bentkus, V. & Götze, F. & Paulauskas, V., 1996. "Bounds for the accuracy of Poissonian approximations of stable laws," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 55-68, December.
    • Handle: RePEc:eee:spapps:v:65:y:1996:i:1:p:55-68
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    References listed on IDEAS

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    1. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook9401, December.
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    2. José Antonio Adell & Alberto Lekuona, 2007. "Berry–Esseen Bounds for Standardized Subordinators via Moduli of Smoothness," Journal of Theoretical Probability, Springer, vol. 20(2), pages 221-235, June.

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