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 < alpha < 2. Let E = {E(t), t >= 0) be the hitting time of a stable subordinator of index 0 < beta < 1 independent of X. We use a connection between X(E(t)) and the stable subordinator of index beta/alpha to derive information on the path behavior of X(E(t)). This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin [Bertoin, J., 1996a. Iterated Brownian motion and stable (1/4) subordinator. Statist. Probab. Lett., 27, 111-114; Bertoin, J., 1996b, Levy Processes. Cambridge University Press]. Using this connection, we obtain various laws ofthe iterated logarithm for X(E(t)). In particular, weestablish the law of the iterated logarithm for local time Brownian motion, X(L(t)), where X is a Brownian motion (the case alpha = 2) and L(t) is the local time at zero of a stable process Y of index 1 < gamma <= 2 independent of X. In this case E(rho t) = L(t) with beta = 1-1/gamma for some constant rho > 0. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper [Meerschaert, M.M., Nane, E., Xiao, Y.. 2008. Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346,432-445]. We also obtain exact small ball probability for X(E(t)) using ideas from Aurzada and Lifshits [Aurzada, F., Lifshits, M., On the small deviation problem for some iterated processes. preprint: arXiv:0806.2559]. (C) 2009 Elsevier B.V. All rights reserved.