General Laws of the Iterated Logarithm for Levy Processes

Let ${X(t) : 0 \leq t < \infty}$ be a real-valued process with stationary independent increments. In this paper, we obtain necesary and sufficint condition for there to exist a positive, nondecreasing function $\beta(t)$ so that $0 < lim sup $\mid$X(t)$\mid$/\beta(t) < \infty$ a.s. both as t tends to zero and infinity. When no such $\beta(t)$ exists we give a simple integral test for whether $lim sup $\mid$X(t)$\mid$/\beta(t)$ is zero or infinity for a given $\beta(t)$.