A Functional Law of the Iterated Logarithm for a Class of Subordinators

Abstract

Let $\{X(t), t \in \lbrack 0, \infty)\}$ be a subordinator whose Levy spectral function $H(x)$ satisfies the inequality $c_1x^{-\alpha} \leq - H(x) \leq c_2x^{-\alpha},$ for all $x > 0$, for a $\alpha \in (0, 1)$ and for certain constants $c_1$ and $c_2, 0 < c_1 \leq c_2 < \infty$. In this paper we obtain (in the $M_1$ topology) the set of all almost sure limit functions of the sequence $(n^{-1/\alpha}X(nt))^{\frac{1}{\log \log n}}, t \in \lbrack 0, 1\rbrack, n \geq 3.$