On the $\gamma$-Variation of Processes with Stationary Independent Increments

Abstract

Let $\{X_t; t \geqq 0\}$ be a stochastic process in $R^N$ defined on the probability space $(\Omega, \mathscr{F}, \mathbf{P})$ which has stationary independent increments. Let $\nu$ be the Levy measure for $X_t$ and let $\beta = \inf\{\alpha > 0: \int_{|x| < 1}|x|^\alpha\nu(dx) < \infty\}$. For each $\omega \in \Omega$, let $V_\gamma(\mathbf{X}(\bullet, \omega); a, b) = \sup \sum^m_{j=1} |X(t_j, \omega) - X(_{t-1}, \omega)|^\gamma$ where the supremum is over all finite subdivisions $a = t_0 < t_1 < \cdots < t_m = b$. Then if $\gamma > \beta, \mathbf{P}\{\mathbf{V}_\gamma(\mathbf{X}(\bullet, \omega); a, b) < \infty\} = 1$.