Variational sums for additive processes

Abstract

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