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

Abstract

Let X1, X2, ... be a sequence of independent and identically distributed (i.i.d.)Rd-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.