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Berry–Esseen Bounds for Standardized Subordinators via Moduli of Smoothness

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  • José Antonio Adell

    (Universidad de Zaragoza)

  • Alberto Lekuona

    (Universidad de Zaragoza)

Abstract

We introduce moduli of smoothness techniques to deal with Berry–Esseen bounds, and illustrate them by considering standardized subordinators with finite variance. Instead of the classical Berry–Esseen smoothing inequality, we give an easy inequality involving the second modulus. Under finite third moment assumptions, such an inequality provides the main term of the approximation with small constants, even asymptotically sharp constants in the lattice case. Under infinite third moment assumptions, we show that the optimal rate of convergence can be simply written in terms of the first modulus of smoothness of an appropriate function, depending on the characteristic random variable of the subordinator. The preceding results are extended to standardized Lévy processes with finite variance.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:2:d:10.1007_s10959-007-0062-5
    DOI: 10.1007/s10959-007-0062-5
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. José A. Adell, 2022. "Probabilistic Stirling Numbers of the Second Kind and Applications," Journal of Theoretical Probability, Springer, vol. 35(1), pages 636-652, March.

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