Dimension of multivariate selfsimilar processes

• Applicant: Professor Dr. Peter Kern
• Subject Area: Mathematics
• Term: from 2013 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 242360936

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.

• DFG Programme: Research Grants