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) =([t/Delta n])Sigma(i=1) f(X-i Delta n-X(i-1)Delta n), where X is a 1-dimensional semimartingale and f a suitable test function, typically f(x) = vertical bar x vertical bar(r), as Delta(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. (C) 2007 Elsevier B.V. All rights reserved.