A law of iterated logarithm for increasing self–similar Markov processes

We consider increasing self–similar Markov processes (Xt, t ≥ 0) on ]0,∞[. By using the Lamperti’s bijection between self–similar Markov processes and Lévy processes, we determine the functions f for which there exists a constant c ∈ R+ \{0} such that lim inft→∞Xt/f(t) = c with probability 1. The determination of such functions depends on the subordinator ξ associated to X through the distribution of the Lévy exponential functional and the Laplace exponent of ξ. We provide an analogous result for the self–similar Markov process associated to the opposite of a subordinator.