  This paper is concerned with asymptotic behavior of a variety of functionals
of increments of continuous semimartingales. Sampling times are assumed to
follow a rather general discretization scheme. If an underlying semimartingale
is thought of as a financial asset price process, a general sampling scheme
like the one employed in this paper is capable of reflecting what happens
whenever the financial trading data are recorded in a tick-by-tick fashion. A
law of large numbers and a central limit theorem are proved after an
appropriate normalization. One application of our result is an extension of the
realized kernel estimator of integrated volatility to the case of random
sampling times.
