engleski

Scaling properties of stochastic processes with applications to parameter estimation and sample path properties

Scaling properties of stochastic processes describe the behavior of the process at different time scales and its distributional properties with respect to aggregation. In the first part of the thesis scaling properties are used to define estimation methods for the tail index of weakly dependent heavy-tailed sequences. In the next step, estimation methods are developed for the linear fractional stable noise as an example of a heavy-tailed model with strong dependence. Scaling is studied by establishing the limiting behavior of the partition function which is a kind of moment statistic for stationary increments stochastic processes. The established results have implications in detecting multifractal processes which are characterized by a nonlinear scaling of the logarithms of their moments in time. In the final part, scaling is related with path properties by establishing bounds on the support of the spectrum of singularities. The thesis introduces new methods in the theory of parameter estimation of models considered that have several advantages over the standard estimators. In addition, it is shown that the standard methods of detecting multifractality are unreliable. Bounds on the support of the spectrum of singularities give a new insight into the questions of regularity of sample paths of stochastic processes.

partition function; scaling function; heavy-tailed distributions; tail index; linear fractional stable motion; Hurst parameter; multifractality; H\"older continuity; spectrum of singularities

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