Journal of Applied Analysis 10 (2004), No. 1, 083--104
Copyright Heldermann Verlag 2004
Random Sums of Independent Random Vectors Attracted by (Semi)-Stable Hemigroups
Peter Becker-Kern
Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany
pbk@math.uni-dortmund.de
[Abstract-pdf]
Let $(X_n)$ be a sequence of independent not necessarily identically distributed random vectors belonging to the domain of attraction of a stable or semistable hemigroup, i.e.~for an increasing sampling sequence $(k_n)$ such that $k_{n+1}/k_n\rightarrow c\geq1$ and linear operators $A_n$, the normalized sums $A_n\sum_{k=\lfloor k_ns\rfloor+1}^{\lfloor k_nt\rfloor}X_k$ converge in distribution uniformly on compact subsets of $\{0\leq s < t\}$ to some full probability $\mu_{s,t}$. Suppose that $(T_n)$ is a sequence of positive integer valued random variables such that $T_n/k_n$ converges in probability to some positive random variable, where we do not assume $(X_n)$ and $(T_n)$ to be independent. Then weak limit theorems of random sums, where the sampling sequence $(k_n)$ is replaced by random sample sizes $(T_n)$, are presented.
Keywords: Random sum, semistable hemigroup, random sample size, Anscombe-condition, operator selfdecomposability, random centering.
MSC: 60F05; 60E07
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