A law of the iterated logarithm for semistable laws on vector spaces and homogeneous groups
This paper is concerned with an iterated logarithm type result for the normed sum of a geometrically increasing number of i.i.d. random vectors \(\{X_i,i=1,2, \dots\}\) in \(d\)-dimensional Euclidean space which have an infinitely divisible law without Gaussian component. It is shown that the result also applies in the case of corresponding distributions on certain homogeneous groups, in particular Heisenberg groups. It is also shown that in the case \(d=1\) the results hold if the \(X_i\) just belong to the domain of normal attraction of the relevant infinitely divisible law. The primary result is new even in the one-dimensional case.
scientific article; zbMATH DE number 1213226
scientific article