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

Consider a semimartingale of the form $Y_t=Y_0+\int_0^ta_sds+\int_0^t\si_{s-}~dW_s$, where $a$ is a locally bounded predictable process and $\si$ (the ``volatility'') is an adapted right--continuous process with left limits and $W$ is a Brownian motion. We define the realised bipower variation process $V(Y;r,s)^n_t=n^{{r+s\over2}-1}\sum_{i=1}^{[nt]} |Y_{i\over n}-Y_{i-1\over n}|^r|Y_{i+1\over n}-Y_{i\over n}|^s$, 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 $\si$ is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with $W$ and by a Poisson random measure, we prove a central limit theorem, in the sense that $\rn~(V(Y;r,s)^n-V(Y;r,s))$ 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 $\si$. We also provide a multivariate version of these results.