Functional Central Limit Theorem for Additive Functionals of α-stable Processes

Abstract

Let β be a standard Brownian motion, let X be an α-stable process, and let $f=\widehat \mu$ be the Fourier transform of a discrete measure. It is shown that weakly in C([0, + ∞ )),

$$ \eta ^{\alpha /2} \int_0^t f(\eta X_s)\text{d}s \Rightarrow \sqrt{C_{f,\alpha}}\beta_t\qquad \text{as $\eta\to +\infty$,} $$

or equivalently

$$ \frac 1 {\sqrt{\lambda}} \int_0^{\lambda t} f(X_s)\text{d}s \Rightarrow \sqrt{C_{f,\alpha}}\beta_t\qquad \text{as $\lambda \to +\infty$.} $$