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

Banach Center Publications (2010)

- Volume: 90, Issue: 1, page 39-65
- ISSN: 0137-6934

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topTomasz J. Kozubowski. "Geometric infinite divisibility, stability, and self-similarity: an overview." Banach Center Publications 90.1 (2010): 39-65. <http://eudml.org/doc/281641>.

@article{TomaszJ2010,

abstract = {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.},

author = {Tomasz J. Kozubowski},

journal = {Banach Center Publications},

keywords = {asymmetric Laplace distribution; fractional Brownian motion; gamma process; geometric distribution; geometric stable distribution; heavy tail; infinite divisibility; Lévy process; linnik distribution; Mittag-Leffler distribution; operator stable distribution; random stability; random sum; scale mixture; self-similarity; stable distribution; subordination},

language = {eng},

number = {1},

pages = {39-65},

title = {Geometric infinite divisibility, stability, and self-similarity: an overview},

url = {http://eudml.org/doc/281641},

volume = {90},

year = {2010},

}

TY - JOUR

AU - Tomasz J. Kozubowski

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

JO - Banach Center Publications

PY - 2010

VL - 90

IS - 1

SP - 39

EP - 65

AB - 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.

LA - eng

KW - asymmetric Laplace distribution; fractional Brownian motion; gamma process; geometric distribution; geometric stable distribution; heavy tail; infinite divisibility; Lévy process; linnik distribution; Mittag-Leffler distribution; operator stable distribution; random stability; random sum; scale mixture; self-similarity; stable distribution; subordination

UR - http://eudml.org/doc/281641

ER -

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