Free-Knot Spline Approximation of Stochastic Processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the $s$-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average $L_p$-distance to the splines spaces, as the (expected) number $k$ of free knots tends to infinity.Comment: 23 page

A first step towards a direct inversion of the Lyman forest in QSO spectra

A method for the recovery of the real space line-of-sight mass density field from Lyman absorption in QSO spectra is presented. The method makes use of a Lucy-type algorithm for the recovery of the HI density. The matter density is inferred from the HI density assuming that the absorption is due to a photoionized intergalactic medium which traces the mass distribution as suggested by recent numerical simulations. Redshift distortions are corrected iteratively from a simultaneous estimate of the peculiar velocity. The method is tested with mock spectra obtained from N-body simulations. The density field is recovered reasonably well up to densities where the absorption features become strongly saturated. The method is an excellent tool to study the density probability distribution and clustering properties of the mass density in the (mildly) non-linear regime. Combined with redshift surveys along QSO sightlines the method will make it possible to relate the clustering of high-redshift galaxies to the clustering of the underlying mass density. We further show that accurate estimates for \Omega_{bar}h^2)^2 J^{-1} H(z)^{-1} and higher order moments of the density probability function can be obtained despite the missing high density tail of the density distribution if a parametric form for the probability distribution of the mass density is assumed.Comment: 10 pages, 6 figure

Financial correlations at ultra-high frequency: theoretical models and empirical estimation

A detailed analysis of correlation between stock returns at high frequency is compared with simple models of random walks. We focus in particular on the dependence of correlations on time scales - the so-called Epps effect. This provides a characterization of stochastic models of stock price returns which is appropriate at very high frequency.Comment: 22 pages, 8 figures, 1 table, version to appear in EPJ

Universality of random matrix dynamics

We discuss the concept of width-to-spacing ratio which plays the central role in the description of local spectral statistics of evolution operators in multiplicative and additive stochastic processes for random matrices. We show that the local spectral properties are highly universal and depend on a single parameter being the width-to-spacing ratio. We discuss duality between the kernel for Dysonian Brownian motion and the kernel for the Lyapunov matrix for the product of Ginibre matrices.Comment: 15 pages, 3 figure