Laws of the iterated logarithm for self-normalised levy processes at zero

We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as t↓0) for the “self-normalised” process {Formula presented}, constructed from a Levy process (X_t)_t≥0 having quadratic variation process (V_t)_t≥0, and an appropriate choice of the constant a. We apply them to obtain LILs when X_t is in the domain of attraction of the normal distribution as t ↓ 0, when X_t is symmetric and in the Feller class at 0, and when X_t is a strictly α-stable process. When X_t is attracted to the normal distribution, an important ingredient in the proof is a Cramer-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.