Multidimensional Stochastic Models and Their Applications

概率统计及其应用的发展导致了模型的日益复杂，涉及到大量的变量，以及包括各种随机因素在内的隐含或显性参数。我们的研究对象包括随机矩阵和随机多项式、高斯场、随机多面体、作为Levy过程生成元的分数拉普拉斯函数、U-max统计量和核密度估计等，这些都是当前研究的热点。这一领域的研究需要以最新的数学成果为基础，利用技术解决现代概率统计中的基本开放问题。特别是，我们将集中解决关于U-max统计量的开放问题的解决，随机多面体和内禀体积与高斯随机矩阵和高斯过程的关系，量化和编码误差的复杂性度量，分数过程的分布以及小偏差问题。

The developments of Probability, Statistics, and their applications lead to increasingly complex models, involving a large number of variables as well as hidden or explicit parameters including a variety of random factors as well. The objects of our research like random matrices and random polynomials, Gaussian fields, random polytopes, fractional Laplacians as generators of Levy processes, U-max statistics and kernel density estimators are topics which are currently intensively studied. Research in this area requires the solution of basic open questions in modern Probability and Statistics using techniques relying on the latest mathematical achievements. In particular we shall concentrate on the solution of open problems concerning U-max statistics, the relation of random polytopes and intrinsic volumes to Gaussian random matrices and Gaussian processes, complexity measures for quantization and coding errors and the distribution of fractional processes as well as problems of small deviations.