Spectral Characterization of the Non-Independent Increment Family of Alpha-Stable Processes that Generalize Gaussian Process Models.

37 Pages Posted: 4 Jan 2017

See all articles by Nourddine Azzaoui

Nourddine Azzaoui

Mathematics Department, Université Blaise Pascal

Gareth Peters

University of California Santa Barbara; University of California, Santa Barbara

Arnaud Guillin

Mathematics Department, Université Blaise Pascal

Malcolm Egan

Université Blaise Pascal (Clermont-Ferrand II)

Date Written: January 2, 2017

Abstract

The characterization of spatial or temporal processes by a family of sufficient functions such as via a unique spectral representation and a mean function forms the basis of a large number of statistical modelling approaches. For instance, in Gaussian process regression modelling, the mean and covariance function specify uniquely the properties of the resulting statistical model. One can therefore parameterize such regression models and understand their structure and attributes generally via a specification of the covariance kernel. In this paper we generalize significantly the class of available spatial and temporal processes that may be used in statistical applications like regressions to allow for non-stationarity and heavy-tailedness. Importantly, we are able to demonstrate for the first time how to achieve this through a novel formulation no longer requiring independence of the increments in the stochastic construction of the process and statistical model.

Furthermore, we achieve this in a manner akin to the covariance kernel specification in a Gaussian process model, we develop a novel covariation spectral representation of some non-stationary and non-indepenent increments symmetric Alpha-stable processes (SalphaS). Such a representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence and should allow a very wide class of statistical regression models to be subsequently developed. We present a general framework for sufficient conditions to characterize such processes and develop general constructive approaches to building models satisfying these conditions.

Suggested Citation

Azzaoui, Nourddine and Peters, Gareth and Guillin, Arnaud and Egan, Malcolm, Spectral Characterization of the Non-Independent Increment Family of Alpha-Stable Processes that Generalize Gaussian Process Models. (January 2, 2017). Available at SSRN: https://ssrn.com/abstract=2892547 or http://dx.doi.org/10.2139/ssrn.2892547

Nourddine Azzaoui

Mathematics Department, Université Blaise Pascal ( email )

24 Avenue des Landais
63117 Aubière Cedex
France

Gareth Peters (Contact Author)

University of California Santa Barbara ( email )

Santa Barbara, CA 93106
United States

University of California, Santa Barbara ( email )

Arnaud Guillin

Mathematics Department, Université Blaise Pascal ( email )

24 Avenue des Landais
63117 Aubière Cedex
France

Malcolm Egan

Université Blaise Pascal (Clermont-Ferrand II) ( email )

24 Avenue des Landais
63117 Aubière Cedex
France

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