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Class L of multivariate distributions and its subclasses

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  • Sato, Ken-iti

Abstract

For any class Q of distributions on Rd, let (Q) be the class of limit distributions of bn-1(X1 + ... + Xn) - an, where {Xn} are independent Rd-valued random variables, each with distribution in Q, bn > 0, an [set membership, variant] Rd, and {bn-1Xj} is a null array. When Q is the class of all distributions on Rd. (Q) = L0 is the usual class L. Define Lm = (Lm-1) and L[infinity] = [intersection]Lm. It is shown that this definition of Lm is equivalent to Urbanik's definition. Description of Lévy measures and representation of characteristic functions of members in these classes are given. Other characterizations of the class L[infinity] are made. Conditions for convergence in terms of the representations are given. Continuity properties of distributions of class L are studied.

Suggested Citation

  • Sato, Ken-iti, 1980. "Class L of multivariate distributions and its subclasses," Journal of Multivariate Analysis, Elsevier, vol. 10(2), pages 207-232, June.
    • Handle: RePEc:eee:jmvana:v:10:y:1980:i:2:p:207-232
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    Citations

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    Cited by:

    1. Pérez-Abreu, Victor & Stelzer, Robert, 2014. "Infinitely divisible multivariate and matrix Gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 155-175.
    2. Chi, Yichun & Yang, Jingping & Qi, Yongcheng, 2009. "Decomposition of a Schur-constant model and its applications," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 398-408, June.
    3. Akita, Koji & Maejima, Makoto, 2002. "On certain self-decomposable self-similar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 53-59, August.
    4. Maejima, Makoto & Sato, Ken-iti & Watanabe, Toshiro, 2000. "Distributions of selfsimilar and semi-selfsimilar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 47(4), pages 395-401, May.
    5. Buchmann, Boris & Lu, Kevin W. & Madan, Dilip B., 2020. "Self-decomposability of weak variance generalised gamma convolutions," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 630-655.
    6. Cohen, Serge & Maejima, Makoto, 2011. "Selfdecomposability of moving average fractional Lévy processes," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1664-1669, November.
    7. Jurek, Zbigniew J., 2023. "Which Urbanik class Lk, do the hyperbolic and the generalized logistic characteristic functions belong to?," Statistics & Probability Letters, Elsevier, vol. 197(C).
    8. Rajba, Teresa, 2009. "Nested classes of C-semi-selfdecomposable distributions," Statistics & Probability Letters, Elsevier, vol. 79(24), pages 2469-2475, December.

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