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Bounds for the accuracy of Poissonian approximations of stable laws

Author

Listed:
  • 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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    Cited by:

    1. Janssen, Arnold, 2000. "Invariance principles for sums of extreme sequential order statistics attracted to Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 255-277, February.
    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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