Grahovac, Danijel (2015) Scaling properties of stochastic processes with applications to parameter estimation and sample path properties. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Abstract
Scaling properties of stochastic processes refer to the behavior of the process at different time scales and distributional properties of its increments with respect to aggregation. In the first part of the thesis, scaling properties are studied in different settings by analyzing the limiting behavior of two statistics: partition function and the empirical scaling function. In Chapter 2 we study asymptotic scaling properties of weakly dependent heavytailed sequences. These results are applied on the problem of estimation of the unknown tail index. The proposed methods are tested against some existing estimators, such as Hill and the moment estimator. In Chapter 3 the same problem is analyzed for the linear fractional stable noise, which is an example of a strongly dependent heavytailed sequence. Estimators will be developed for the Hurst parameter and stable index, the main parameters of the linear fractional stable motion. Chapter 4 contains an overview of the theory of multifractal processes, which can be characterized in several di erent ways. A practical problem of detecting multifractal properties of time series is discussed from the point of view of the results of the preceding chapters. The last Chapter 5 deals with the fine scale properties of the sample paths described with the socalled spectrum of singularities. The new results are given relating scaling properties with path properties and applied to different classes of stochastic processes.
Item Type:  Thesis (Doctoral thesis) 

Keywords:  partition function, scaling function, heavytailed distributions, tail index, linear fractional stable motion, Hurst parameter, multifractality, Hölder continuity, spectrum of singularities 
Supervisor:  Leonenko, Nikolai N. and Benšić, Mirta 
Date:  2015 
Number of Pages:  112 
Subjects:  NATURAL SCIENCES > Mathematics > Probability Theory and Statistics 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  27 Apr 2015 12:24 
Last Modified:  27 Apr 2015 13:13 
URI:  http://digre.pmf.unizg.hr/id/eprint/3885 
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