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(E(t)). This is an extension of the connection of iterated Brownian motion and (14)-stable subordinator due to Bertoin [Bertoin, J., 1996a. Iterated Brownian motion and stable (14) subordinator. Statist. Probab. Lett., 27, 111-114; Bertoin, J., 1996b, Lévy Processes. Cambridge University Press]. Using this connection, we obtain various laws of the iterated logarithm for X(E(t)). In particular, we establish the 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 [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].