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Topics in Infinitely Divisible Distributions and Lévy Processes, Revised Edition pp 1–26Cite as

Classes Lm and Their Characterization

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Abstract

Selfdecomposable distributions are extensions of stable distributions and form a subclass of the class of infinitely divisible distributions. In this chapter we will introduce, between the class L 0 of selfdecomposable distributions and the class \(\mathfrak {S}\) of stable distributions, a chain of subclasses called L m, m = 1, …, ∞:

$$\displaystyle L_{0}\supset L_{1}\supset L_{2}\supset \cdots \supset L_{\infty }\supset \mathfrak {S}. $$

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Notes

  1. 1.

    In Loève’s book infinitely divisible distributions are named as infinitely decomposable.

  2. 2.

    The property (1.1) for Z nk is called infinitesimal by Gnedenko and Kolmogorov [26] (1968) and uniformly asymptotically negligible by Loève [58] (1977, 1978). Chung [19] (1974) uses the word holospoudic double array. We follow Feller [24] (1971) in using the word “null array”.

  3. 3.

    The proof of Proposition 1.22 uses Lévy– Khintchine representation in Theorem 1.28. But Theorem 1.28 is shown independently of the theory of stable distributions.

  4. 4.

    This definition of α-stability and strict α-stability is slightly different from that of [93], where trivial distributions do not have index in stability and δ 0 does not have index in strict stability. Thus neither (1.9) nor (1.10) is true if \(\mathfrak {S}_{\alpha }\) and \(\mathfrak {S}_{\alpha }^0\) are, respectively, the classes of α-stable and strictly α-stable distributions in the sense of [93].

  5. 5.

    We say that a statement S(ω) involving ω is true almost surely (or a.s.) if there is \(\Omega _0\in \mathcal {F}\) with P[ Ω0] = 1 such that S(ω) is true for all ω ∈ Ω0.

  6. 6.

    Cone-parameter Lévy processes in Chap. 4 do not have these properties (see Example 4.36).

  7. 7.

    If \(\mu \in \mathfrak {S}_{\alpha }\) with 1 < α < 2, then ∫|x|>1|x|ν(dx) < ∞, which is shown by (1.40). For any μ ∈ ID, ∫|x|>1|x|ν(dx) < ∞ and \(\int _{\mathbb {R}^d} |x|\mu (dx)<\infty \) are equivalent and, in this case, \(\int _{\mathbb {R}^d} x\mu (dx)=\gamma +\int _{|x|>1} x\nu (dx)\) ([93] Example 25.12).

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Authors and Affiliations

  1. Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Sinaloa, Culiacán, Mexico

    Alfonso Rocha-Arteaga

  2. Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, Japan

    Ken-iti Sato

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  1. Alfonso Rocha-Arteaga
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  2. Ken-iti Sato
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Rocha-Arteaga, A., Sato, Ki. (2019). Classes Lm and Their Characterization. In: Topics in Infinitely Divisible Distributions and Lévy Processes, Revised Edition. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-22700-5_1

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