Geometric Compound Invariance Principle Linnik Distribution Mittag-Leffler Distribution Random Sum Stable Distribution Stochastic Integral
Issue Date:
1999
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 25, No 3, (1999), 241p-256p
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.
