Random sums of independent random vectors attracted by (semi)-stable hemigroups. (English) Zbl 1109.60014
Summary: 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 \(k_{n+1}/ k_n\to c\geq 1\) and linear operators \(A_n\), the normalized sums \(A_n\sum^{\lfloor k_nt\rfloor}_{k=\lfloor k_ns \rfloor+1}X_k\) 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.
MSC:
| 60F05 | Central limit and other weak theorems |
| 60E07 | Infinitely divisible distributions; stable distributions |
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