Dimension of multivariate selfsimilar processes

自相似随机过程的特点是在时空尺度变换下其分布不变性。在许多领域中，这些过程已经证明了它们对具有自相似特性的随机动力学行为建模的重要性。对于多变量自相似过程，线性算子的空间尺度提供了最高程度的建模灵活性。此外，对于自相似随机场，可以在时域中引入算子缩放。在这些情况下，标度算子的特征在于指数本身是线性算子。研究了算子自相似过程和算子自相似随机场的维数结果。作为以前结果的一个重要推广，我们的目标是给出算子自相似过程和算子自相似平稳增量随机场样本路径的Hausdorff维数的一个完整刻画。由于Hausdorff维数被很好地理解为粗糙度的度量，因此所提出的结果提供了关于所研究的过程类内的样本路径的局部行为的有价值的信息。属于这类过程的重要例子是算子分数布朗运动。为了更深入地研究局部样本路径行为的精细结构，我们进一步给出了算子自相似Lévy过程的精确Hausdorff测度函数。所提出的维数公式只依赖于自相似指数的特征值的真实的部分，即使对于算子半自相似的较弱标度性质也可能成立。从应用的角度看，特征值的真实的部分的有效估计过程是非常重要的。该项目的另一个目的是扩展这些参数的一致估计程序，以适用于运营商自相似随机场。目前，该方法对一般算子半自相似过程是有效的，我们的目标是进一步给出收敛速度的结果。

Selfsimilar stochastic processes are characterized by invariance of their distributions under space-time scaling transformations. In many areas these processes have proved their importance for modeling random dynamical behavior with selfsimilarity properties. For multivariate selfsimilar processes a spacial scaling with linear operators provides the highest degree of flexibility in modeling. Additionally, for selfsimilar random fields an operator scaling can be introduced in the time domain. In these cases the scaling operators are characterized by an exponent which is itself a linear operator. The research project is concerned with dimension results for operator selfsimilar processes and operator selfsimilar random fields. As a significant generalization of previous results, we aim to give a full characterization of the Hausdorff dimension for sample paths of operator selfsimilar processes and operator selfsimilar random fields with stationary increments. Since the Hausdorff dimension is well understood as a measure of roughness, the proposed results give valuable information on the local behavior of sample paths within the studied class of processes. Important examples of processes belonging to this class are operator fractional Brownian motions. To go even more into the fine structure of local sample path behavior, we further aim to give exact Hausdorff measure functions for operator selfsimilar Lévy processes. The proposed dimension formulas do only depend on the real parts of the eigenvalues of the selfsimilarity exponent and are likely to hold even for the weaker scaling property of operator semi-selfsimilarity. In view of applications, an efficient estimation procedure for the real parts of the eigenvalues is of great importance. A further aim of the project is to extend a consistent estimation procedure for these parameters to be applicable for operator selfsimilar random fields. At present, the procedure is valid for general operator semi-selfsimilar processes and we further aim to give rate of convergence results.