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

Abstract

We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as t ↓ 0 t\downarrow 0 ) for the “self-normalised” process ( X t − a t ) / V t (X_{t}-at)/\sqrt {V_{t}} , t > 0 t>0 , constructed from a Lévy process ( X t ) t ≥ 0 (X_{t})_{t\geq 0} having quadratic variation process ( V t ) t ≥ 0 (V_{t})_{t\geq 0} , and an appropriate choice of the constant a a . We apply them to obtain LILs when X t X_{t} is in the domain of attraction of the normal distribution as t ↓ 0 t\downarrow 0 , when X t X_{t} is symmetric and in the Feller class at 0, and when X t X_{t} is a strictly α − \alpha - stable process. When X t X_{t} is attracted to the normal distribution, an important ingredient in the proof is a Cramér-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.