**A Central Limit
Theorem for Realised Power and Bipower Variations of Continuous
Semimartingales**

**Ole BARNDORFF--NIELSEN**,

Department of Mathematical Sciences,

University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

**Svend Erik GRAVERSEN**,

Department of Mathematical Sciences,

University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

**Jean JACOD**,

Laboratoire de Probabilites et

Modeles Aleatoires (CNRS UMR 7599)

Universite P. et M. Curie, 4 Place Jussieu,

75 252 - Paris Cedex,

France.

**Mark PODOLSKIJ**,

Dept. of Probability and Statistics,

Ruhr University of Bochum,

Universitatstrasse 150, 44801 Bochum, Germany,

**Neil SHEPHARD**,

Nuffield College, Oxford OX1 1NF, UK.

**Abstract**

Consider a semimartingale of the form
Y_{t}=Y_0+\int _0^{t}a_{s}ds+\int _0^{t}_{s-} dW_{s}, where a is a locally
bounded predictable process and (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)_{t}^n=n^{((r+s)/2)-1}
\sum_{i=1}^{[nt]}|Y_{(i/n)}-Y_{((i-1)/n)}|^{r}|Y_{((i+1)/n)}-Y_{(i/n)}|^{s},
where r and s are nonnegative reals with r+s>0. We prove that
V(Y;r,s)_{t}n converges locally uniformly in time, in probability, to a
limiting process V(Y;r,s)_{t} (the "bipower variation process"). If
further 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 \sqrt(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 \sigma. We also provide a multivariate version of
these results.

**Keywords:** Central limit
theorem, quadratic variation, bipower variation,

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