Large deviations for local time fractional Brownian motion and applications

Let $W^H=\{W^H(t),t\in \text{rr}\}$ be a fractional Brownian motion of Hurst index $H\in (0,1)$ with values in $\text{rr}$, and let $L=\{L_t,t\ge 0\}$ be the local time process at zero of a strictly stable L'evy process $X={X_t, t \ge 0}$ofindex$1<\alpha\leq 2$independentof$W^H$\mathrm{.}The$\a$-stable local time fractional Brownian motion $Z^H=\{Z^H(t),t\ge 0\}$ is defined by $Z^H(t) = W^H(L_t)$\mathrm{.}Theprocess$Z^H$isself-similarwithself-similarityindex$H(1

• \frac 1 \alpha)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coupleCTRW,limitCTRW}). 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 $Z^H$.