Discrete alloy-type models: Regularity of distributions and recent results

We consider discrete random Schr\"odinger operators on $\ell^2 (\mathbb{Z}^d)$ with a potential of discrete alloy-type structure. That is, the potential at lattice site $x \in \mathbb{Z}^d$ is given by a linear combination of independent identically distributed random variables, possibly with sign-changing coefficients. In a first part we show that the discrete alloy-type model is not uniformly $\tau$-H\"older continuous, a frequently used condition in the literature of Anderson-type models with general random potentials. In a second part we review recent results on regularity properties of spectral data and localization properties for the discrete alloy-type model.Comment: 20 pages, 0 figure

Effects of Macroeconomic Shocks to the Quality of the Aggregate Loan Portfolio

The paper concerns macro-prudential analysis. It uses an unrestricted VAR model to empirically investigate transmission involving a set of macroeconomic variables describing the development of the Czech economy and the functioning of its credit channel in the past eleven years. Its novelty lies in the fact that it provides the first systematic assessment of the links between loan quality and macroeconomic shocks in the Czech context. The VAR methodology is applied to monthly data transformed into percentage change. The out-of-sample forecast indicates that the most likely outlook for the quality of the banking sector's loan portfolio is that up to the end of 2006 the share of non-performing loans in it will follow a slightly downward trend below double-digit rates. The impulse response is augmented by stress testing exercises that enable us to determine a macroeconomic early warning signal of any worsening in the quality of banks' loans. The paper suggests that the Czech banking sector has attained a considerable ability to withstand a credit risk shock and that the banking sector's stability is compatible both with price stability and with economic growth. Despite being devoted to empirical investigation, the paper pays great attention to methodological issues. At the same time it tries to present both the VAR model and its results transparently and to openly discuss their weak points, which to a large degree can be attributed to data constraints or to the evolutionary nature of an economy in transition.Czech Republic, Macro-prudential analysis, Non-performing loans, VAR model.

Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators

We discuss properties of $L^2$-eigenfunctions of Schr\"odinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.Comment: Keywords: scale-free unique continuation property, equidistribution property, observability estimate, uncertainty relation, Carleman estimate, Schr\"odinger operator, elliptic differential equatio

Spectral diagonal ensemble Kalman filters

A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the aproximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields, which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles.Comment: 15 pages, 4 figure