# Geometric Stable Laws Through Series Representations

Kozubowski, Tomasz; Podgórski, Krzysztof

Serdica Mathematical Journal (1999)

- Volume: 25, Issue: 3, page 241-256
- ISSN: 1310-6600

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topKozubowski, Tomasz, and Podgórski, Krzysztof. "Geometric Stable Laws Through Series Representations." Serdica Mathematical Journal 25.3 (1999): 241-256. <http://eudml.org/doc/11516>.

@article{Kozubowski1999,

abstract = {Let (Xi ) be a sequence of i.i.d. random variables, and let
N be a geometric random variable independent of (Xi ). Geometric stable
distributions are weak limits of (normalized) geometric compounds, SN =
X1 + · · · + XN , when the mean of N converges to infinity. By an appropriate
representation of the individual summands in SN we obtain series
representation of the limiting geometric stable distribution. In addition, we
study the asymptotic behavior of the partial sum process SN (t) = ⅀( i=1 ... [N t] ) Xi ,
and derive series representations of the limiting geometric stable process
and the corresponding stochastic integral. We also obtain strong invariance
principles for stable and geometric stable laws.},

author = {Kozubowski, Tomasz, Podgórski, Krzysztof},

journal = {Serdica Mathematical Journal},

keywords = {Geometric Compound; Invariance Principle; Linnik Distribution; Mittag-Leffler Distribution; Random Sum; Stable Distribution; Stochastic Integral; geometric compound; random sums; stable distributions},

language = {eng},

number = {3},

pages = {241-256},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Geometric Stable Laws Through Series Representations},

url = {http://eudml.org/doc/11516},

volume = {25},

year = {1999},

}

TY - JOUR

AU - Kozubowski, Tomasz

AU - Podgórski, Krzysztof

TI - Geometric Stable Laws Through Series Representations

JO - Serdica Mathematical Journal

PY - 1999

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 25

IS - 3

SP - 241

EP - 256

AB - Let (Xi ) be a sequence of i.i.d. random variables, and let
N be a geometric random variable independent of (Xi ). Geometric stable
distributions are weak limits of (normalized) geometric compounds, SN =
X1 + · · · + XN , when the mean of N converges to infinity. By an appropriate
representation of the individual summands in SN we obtain series
representation of the limiting geometric stable distribution. In addition, we
study the asymptotic behavior of the partial sum process SN (t) = ⅀( i=1 ... [N t] ) Xi ,
and derive series representations of the limiting geometric stable process
and the corresponding stochastic integral. We also obtain strong invariance
principles for stable and geometric stable laws.

LA - eng

KW - Geometric Compound; Invariance Principle; Linnik Distribution; Mittag-Leffler Distribution; Random Sum; Stable Distribution; Stochastic Integral; geometric compound; random sums; stable distributions

UR - http://eudml.org/doc/11516

ER -

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