Law of the iterated logarithm for the increments of stable subordinators (Q1103957)

Let X(t), \(t\in [0,\infty)\), be a stable subordinator of index \(\alpha\), \(0<\alpha <1\). Let \(0<a_ t\leq t\), \(t>0\) be a non-negative function and define \[ y(t)=X(t+a_ t)-X(t),\quad t>0. \] The principal result of this paper is that if \(a_ t\to \infty\) as \(t\to \infty\) and \(t^{- 1}a_ t\) is non-increasing, then \[ \lim_{t\to \infty}\inf y(t)/\beta (t)=1\quad a.s. \] Here \(\beta (t)=\theta_{\alpha}a_ t^{1/\alpha}(\log (t/a_ t)+\log \log t)^{(\alpha -1)/\alpha}\) for a suitable constant \(\theta_{\alpha}\). The proof is a suitable modification of the standard proof of the law of the iterated logarithm. Partial results are obtained on the almost sure behaviour of \[ \lim_{d\to \infty}\sup (\min \{t: y(t)>d\}/\gamma (d)) \] where \(\gamma\) (d) is the inverse function to \(\beta\) (t).