A central limit theorem for realised power and bipower variations of continuous semimartingales

Consider a semimartingale of the form Y-t = Y-0 + integral(t)(0) a(s)ds + integral(t)(0) sigma(s-) dW(s), where a is a locally bounded predictable process and or (the "volatility") is an adapted right-continuous process with left limits and W is a Brownian motion. We consider the realised bipower variation process [GRAPHICS] where r and s are nonnegative reals with r + s > 0. We prove that V(Y;r,s)(n)(t) converges locally uniformly in time, in probability, to a limiting process V(Y;r,s)(t) (the "bipower variation process"). If further or is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove that root n (V(Y;r,s)(n) - V(Y;r,s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W', which is independent of the driving terms of Y and a. We also provide a [SIC].