Asymptotic properties of realized power variations and related functionals of semimartingales

This paper is concerned with the asymptotic behavior of sums of the form U n ( f ) t = ∑ i = 1 [ t / Δ n ] f ( X i Δ n − X ( i − 1 ) Δ n ) , where X is a 1-dimensional semimartingale and f a suitable test function, typically f ( x ) = | x | r , as Δ n → 0 . We prove a variety of “laws of large numbers”, that is convergence in probability of U n ( f ) t , sometimes after normalization. We also exhibit in many cases the rate of convergence, as well as associated central limit theorems.