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 $t\downarrow 0$) for the “self-normalised” process $(X_{t}-at)/\sqrt {V_{t}}$, $t>0$, constructed from a Lévy process $(X_{t})_{t\geq 0}$ having quadratic variation process $(V_{t})_{t\geq 0}$, 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 $t\downarrow 0$, when $X_{t}$ is symmetric and in the Feller class at 0, and when $X_{t}$ is a strictly $\alpha -$stable process. When $X_{t}$ 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.