Laws of the Iterated Logarithm for a Class of Iterated Processes

Let X = {X(t), t ≥ 0} be a Brownian motion or a spectrally negative stable process of index 1 < α < 2. Let E = {E(t), t ≥ 0} be the hitting time of a stable subordinator of index 0 < β < 1 independent of X . We use a connection between X(E(t)) and the stable subordinator of index β/α to derive information on the path behavior of X(Et). This is an extension of the connection of iterated Brownian motion and ( 4 )-stable subordinator due to Bertoin [7]. Using this connection, we obtain various laws of the iterated logarithm for X(E(t)). In particular, we establish law of the iterated logarithm for local time Brownian motion, X(L(t)), where X is a Brownian motion (the case α = 2) and L(t) is the local time at zero of a stable process Y of index 1 < γ ≤ 2 independent of X . In this case E(ρt) = L(t) with β = 1− 1/γ for some constant ρ > 0. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper [27]. We also obtain exact small ball probability for X(Et) using ideas from [2].