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A law of the iterated logarithm for semistable laws on vector spaces and homogeneous groups

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Abstract

Let X1, X2, ... be a sequence of independent and identically distributed (i.i.d.)R d-valued random vectors distributed according to a full (B,c) semistable law without Gaussian component. Then the following law of the iterated logarithm holds.

$$\mathop {\lim \sup }\limits_{n \to \infty } \left\| {B^{ - (n + [\log n/\log c])} \sum\limits_{i = 1}^{[c^n ]} {X_i } } \right\|^{1/\log n} = 1 a.s.$$

This result is new even in the one-dimensional situation of semistable laws on the real line, where we extend our result to laws in the domain of normal attraction of a semistable law. Furthermore, we prove that this kind of law of the iterated logarithm also holds for certain semistable laws on homogeneous groups, especially on Heisenberg groups.

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Authors and Affiliations

  1. Mathematical Institute, University of Dortmund, 44221, Dortmund, Germany

    H. -P. Scheffler

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  1. H. -P. Scheffler
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Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.

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Scheffler, H.P. A law of the iterated logarithm for semistable laws on vector spaces and homogeneous groups. J Math Sci 89, 1545–1552 (1998). https://doi.org/10.1007/BF02362290

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