In this paper, we show that the moving average and series representations of fractional Brownian motion can be obtained using the spectral theory of vibrating strings. The representations are shown to be consequences of general theorems valid for a large class of second-order processes with stationary increments. Specifically, we use the 1-1 relation discovered by M.G. Krein between spectral measures of continuous second-order processes with stationary increments and differential equations describing the vibrations of a string with a certain length and mass distribution. r Zanten), pzareba@cs.vu.nl (P. Zareba). complicated, since it is neither a Markov process nor a semimartingale (unless the Hurst index equals 1 2 ). Representations of the fBm in terms of simpler, better understood processes are therefore of great importance. A key result in this respect is the moving average representationas a stochastic integral of a deterministic kernel w t with respect to an ordinary Brownian motion W. See for instance Molchan [19], Decreusefond and Ü stu¨nel [6], Norros et al. [21], Nuzman and Poor [23] or Pipiras and Taqqu [24] for various proofs of this result. The moving average representation allows us to obtain results regarding for instance prediction, absolute continuity, maximal inequalities, stochastic calculus, etc. (cf. e.g. [6,21,22,18,24,27], to mention but a few). More generally, it allows us to apply the results and techniques available for general Volterra processes to the fBm (see e.g. [2,3,13] for such results).In this paper, we present an approach to (1.1) which has remained unexplored so far. The method also yields a new proof of the series representation of the fBm obtained recently in Dzhaparidze and Van Zanten [11]. The presented approach is not just yet another ad hoc method applicable only to the fBm. On the contrary, the purpose of the paper is to show that the moving average and series representations of the fBm are in fact special cases of results for a much wider class of second-order processes with stationary increments (si-processes). Since there is an increasing interest in general Gaussian si-processes as building blocks for models (cf. e.g. [1,15,17,4]), it seems quite relevant to explore methods that allow us to understand the structure of such processes better.The approach we take was pioneered by M.G. Krein in the 1950s. He investigated problems like interpolation and prediction for stationary processes. The central observation of Krein was that there is a 1-1 relationship between symmetric, Borel measures m on the line satisfying Z mðdlÞ 1 þ l 2 o1 (1.2) and differential operators of the form f 7 !df 0 =dm associated with a vibrating string with mass distribution m. Roughly speaking, the measure m associated with a string describes the kinetic energy of the string as it vibrates at different frequencies. When combined with the theory of reproducing kernel Hilbert spaces (RKHSs) of entire functions of de Branges [5], very precise results are obtained concerning the structure of (subspace...

In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.

We present series expansions and moving average representations of isotropic Gaussian random fields with homogeneous increments, making use of concepts of the theory of vibrating strings. We illustrate our results using the example of Lévy's fractional Brownian motion onℝN.