Large deviations for local time fractional Brownian motion and applications

Let WH = {WH (t), t ∈ R} be a fractional Brownian motion of Hurst index H ∈ (0, 1) with values in R, and let L = {Lt , t 0} be the local time process at zero of a strictly stable Lévy process X = {Xt , t 0} of index 1 < α 2 independent of WH. The α-stable local time fractional Brownian motion Z H = {Z H (t), t 0} is defined by Z H (t) = WH (Lt ). The process Z H is self-similar with self-similarity index H(1 − 1 α ) and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps [P. Becker-Kern, M.M. Meerschaert, H.P. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004) 730–756; M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004) 623–638]. However, Z H does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process Z H . As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for Z H .