A maximal law of the iterated logarithm for operator-normalized stochastically compact partial sums of i.i.d. random vectors

A law of the iterated logarithm for stable laws on homogeneous groups

“…This is the analogue of the known law of the iterated logarithm (L.I.L) for operator stable laws on finite dimensional vector spaces proved in [17,Chapter 5,Theorem]. Recently Neuenschwander and the author [14] proved a related result for the center part of stable measures on the Heisenberg group.…”

“…In [2] it is proved a version of Ottaviani's maximal inequality on measurable groups. We will prove another Ottaviani type maximal inequality and a maximal inequality analogues to [4] and then follow the method of proof presented in [3] and [17].…”

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

“…It is also interesting to compare the following result with the stable ease considered in [18]. It is also interesting to compare the following result with the stable ease considered in [18].…”

“…It follows from the theorem in [12] Since log log[c, d ~ log n as n ---+ oo and since in the case of operator semistable laws one considers the subsequence S[~,] of S,~, it turns out that this is the appropriate generalization of the corresponding result of [18] for operator stable laws. It follows from the theorem in [12] Since log log[c, d ~ log n as n ---+ oo and since in the case of operator semistable laws one considers the subsequence S[~,] of S,~, it turns out that this is the appropriate generalization of the corresponding result of [18] for operator stable laws.…”

“…For the proof of this theorem we combine the techniques of [2] and [18] (see also [17]). It is a remarkable fact that in contrast to the stable case considered in these papers, we need no maximal inequality in the proof of the upper bound.…”