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A law of the iterated logarithm for randomly stopped sums of heavy tailed random vectors. (English) Zbl 0964.60034

Authors’ abstract: Let \(X_1\), \(X_2\), be an i.i.d. sequence of \(\mathbb{R}^d\)-valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that \((T_n)\) is a sequence of positive integer valued random variables such that for some \(\delta_0> 0\) \[ (\log n)^{\delta_0} \Biggl|{T_n\over n}- D\Biggr|\to 0\quad\text{a.s.} \] for some discrete positive random variable \(D\), where we do not assume that \((X_n)\) and \((T_n)\) are independent. Let \(S_n= \sum^n_{i=1} X_i\). Then various laws of the iterated logarithm for the norm of \((S_{T_n})\) as well as the radial projection \(\langle S_{T_n},\theta\rangle\) onto a unit vector \(\theta\) are presented.

MSC:

60F15 Strong limit theorems
60E07 Infinitely divisible distributions; stable distributions
60F05 Central limit and other weak theorems
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