We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process betweenthe successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/orcentering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f. As for the associated central limit theorem,one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.

In this paper we review some recent work on limit results on realised power variation, that is, sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.

The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A0 of some ‘scanning set' A0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A0. We ask if the set A0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A0 from realisations of the sample paths of the random field Z.