Variational sums for additive processes

Abstract:

Let $X(t),0 \leqq t \leqq T$, be an additive process, and let ${X_{nk}}$ be the $k$th increment of $X(t)$ associated with the partition ${\Pi _n}$ of $[0,T]$. Assume $||{\Pi _n}|| \to 0$. Let $\beta$ be the Blumenthal-Getoor index of $X(T)$ and let $2 \geqq \gamma > \beta$. When the partitions are nested, $\sum \nolimits _k {|{X_{nk}}{|^\gamma }}$ converges a.s. to $\sum {\{ |J(s){|^\gamma }:0 \leqq s \leqq T\} }$, where $J(s)$ is the jump of $X(t)$ at $s$. This convergence also holds when the partitions are not nested provided either $X(t)$ has stationary increments or $1 \geqq \gamma > \beta$. This extends a result of P. W. Millar and completes a result of S. M. Berman.