The variation of a stable path is stable

SummaryLet X(t) be a separable symmetric stable process of index α. Let P be a finite partition of [0,1], and ℘ a collection of partitions. The variation of a path X(t) is defined in three ways in terms of the sum $$\sum\limits_{t_i \in P} {|X(t_i ) - X(t_{i - 1} )|^\beta } $$ collection ℘. Under certain conditions on ℘ and on the parameters α and Β, the distribution of the variation is shown to be a stable law. Under other conditions the distribution of the variational sum converges to a stable distribution.