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

Department of Actuarial Mathematics and Statistics, Heriot-Watt University; University College London - Department of Statistical Science; University of Oxford - Oxford-Man Institute of Quantitative Finance; London School of Economics & Political Science (LSE) - Systemic Risk Centre; University of New South Wales (UNSW) - Faculty of Science

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)

Department of Actuarial Mathematics and Statistics, Heriot-Watt University ( email )

Edinburgh Campus
Edinburgh, EH14 4AS
United Kingdom

HOME PAGE: http://garethpeters78.wixsite.com/garethwpeters

University College London - Department of Statistical Science ( email )

1-19 Torrington Place
London, WC1 7HB
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

University of Oxford Eagle House
Walton Well Road
Oxford, OX2 6ED
United Kingdom

London School of Economics & Political Science (LSE) - Systemic Risk Centre ( email )

Houghton St
London
United Kingdom

University of New South Wales (UNSW) - Faculty of Science ( email )

Australia

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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