Inference for the Levy models and their applications in medicine and statistical physics

Levy processes, that is stochastic processes with time-homogeneous and independent increments, found numerous applications in the physical sciences, economics and engineering.; In this dissertation we study specific theoretical issues related to the multiscaling properties of some special classes of Levy processes and to the kinetic equations describing time-evolution of statistical mechanical systems driven by certain Levy processes displaying, perhaps limited, fractal behavior.; To be able to apply these models to real data we also develop statistical parametric estimation procedures for them. These theoretical tools are then utilized in analysis of EEG recordings for fullterm and preterm neonates. The issues of sleep stage separations and long-memory property have been also investigated for this data set.