Geometric infinite divisibility, stability, and self-similarity: an overview

• Autorzy: Tomasz J. Kozubowski
• Języki publikacji: EN

Abstrakty

EN

The concepts of geometric infinite divisibility and stability extend the classical properties of infinite divisibility and stability to geometric convolutions. In this setting, a random variable X is geometrically infinitely divisible if it can be expressed as a random sum of $N_p$ components for each p ∈ (0,1), where $N_p$ is a geometric random variable with mean 1/p, independent of the components. If the components have the same distribution as that of a rescaled X, then X is (strictly) geometric stable. This leads to broad classes of probability distributions closely connected with their classical counterparts. We review fundamental properties of these distributions and discuss further extensions connected with geometric sums, including multivariate and operator geometric stability, discrete analogs, and geometric self-similarity.

Kategorie tematyczne

• 62E10: Characterization and structure theory
• 62E15: Exact distribution theory
• 60G50: Sums of independent random variables; random walks
• 62H05: Characterization and structure theory
• 60G52: Stable processes
• 60-02: Research exposition (monographs, survey articles)
• 60G51: Processes with independent increments; L\'evy processes
• 60F05: Central limit and other weak theorems
• 60E05: Distributions: general theory
• 60G18: Self-similar processes
• 60E07: Infinitely divisible distributions; stable distributions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 90
• Numer: 1
• Strony: 39-65

Daty

wydano

2010

Twórcy

autor

Tomasz J. Kozubowski

• Department of Mathematics & Statistics, University of Nevada, Reno, NV 89523, USA

Identyfikatory

DOI

10.4064/bc90-0-3