Asymptotic properties of realized power variations and related functionals of semimartingales

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