Geometric Stable Laws Through Series Representations

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

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Abstract

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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.

How to cite

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Kozubowski, 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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