Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.     

Almost everywhere convergence of the spherical partial sums for radial functions

LetfL ^p( ⁿ ),n2, be a radial function and letS _Rf be the spherical partial sums operator. We prove that if thenS _Rf(x)f(x) a.e. asR. The result is false for and \frac{{2n}}{{n + 1}}$$ " align="middle" border="0"> .Partially supported by M.P.I.     

Gas-Chromatographie

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Milch

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Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices

A new fabrication procedure for integration of ultraviolet transparent pure-silica planar waveguides, fiber couplers and high-aspect ratio submicrometer channels is presented. Only a single photolithographic mask step is required. The channels are 80-90 microm deep and the width can be reduced to about 0.5 microm, corresponding to a height-to-width ratio of more than 150. The core of the waveguides consists of pure silicon dioxide, which is favorable over doped silica, due to the absence of absorption centers associated with the dopants. This furthermore improves the long-term stability of the waveguides, because of an increased radiation resistance of the glass. The propagation loss decreases from 1.0 dB/cm at 200 nm to 0.2 dB/cm at 800 nm, which, to our knowledge, is the lowest propagation loss reported for integrated planar waveguides in the ultraviolet wavelength region to date. The effective optical path length is 1.2 mm for an absorbance cell with a nominal length of 1.0 mm, indicating effective suppression of stray light. The limit of detection for paracetamol when present in the entire channel network was determined to 3 microg/mL. Finally, the applicability of the fabricated devices for capillary electrophoresis was evaluated by separation of caffein, paracetamol and ketoprofone using absorbance detection at 254 nm.