  This paper is concerned with the asymptotic behavior of sums of terms which
are a test function f evaluated at successive increments of a discretely
sampled semimartingale. Typically the test function is a power function (when
the power is 2 we get the realized quadratic variation) . We prove a variety of
``laws of large numbers'', that is convergence in probability of these sums,
sometimes after normalization. We also exhibit in many cases the rate of
convergence, as well as associated central limit theorems.
