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Spectral Decomposition for Generalized Domains of Semistable Attraction

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  • Mark M. Meerschaert
  • Hans-Peter Scheffler

Abstract

Suppose X, X 1 , X 2 , X 3 ,... are i.i.d. random vectors, and k n a sequence of positive integers tending to infinity in such a way that k n+1 /k n →c≥1. If there exist linear operators A n and constant vectors b n such that $$A_n (X_1 + \cdots + X_{k_n } ) - b_n $$ converges in law to some full limit, then we say that the distribution of X belongs to the generalized domain of semistable attraction of that limit law. The main result of this paper is a decomposition theorem for the norming operators A n , which allows us to reduce the problem to the case where the tail behavior of the limit law is essentially uniform in all radial directions. Applications include a complete description of moments, tails, centering, and convergence criteria.

Suggested Citation

  • Mark M. Meerschaert & Hans-Peter Scheffler, 1997. "Spectral Decomposition for Generalized Domains of Semistable Attraction," Journal of Theoretical Probability, Springer, vol. 10(1), pages 51-71, January.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:1:d:10.1023_a:1022638213920
    DOI: 10.1023/A:1022638213920
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    References listed on IDEAS

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    1. Meerschaert, Mark M., 1993. "Regular variation and generalized domains of attraction in k," Statistics & Probability Letters, Elsevier, vol. 18(3), pages 233-239, October.
    2. Scheffler, H. P., 1994. "Domains of Semi-Stable Attraction of Nonnormal Semi-Stable Laws," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 432-444, November.
    3. Hudson, William N. & Veeh, Jerry Alan & Weiner, Daniel Charles, 1988. "Moments of distributions attracted to operator-stable laws," Journal of Multivariate Analysis, Elsevier, vol. 24(1), pages 1-10, January.
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    Cited by:

    1. Peter Becker-Kern, 2003. "Stable and Semistable Hemigroups: Domains of Attraction and Self-Decomposability," Journal of Theoretical Probability, Springer, vol. 16(3), pages 573-598, July.

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