Abstract
We introduce rank-k-continuous axis-aligned p-generalized elliptically contoured distributions and study their properties such as stochastic representations, moments, and density-like representations. Applying the Kolmogorov existence theorem, we prove the existence of random processes having axis-aligned p-generalized elliptically contoured finite dimensional distributions with arbitrary location and scale functions and a consistent sequence of density generators of p-generalized spherical invariant distributions. Particularly, we consider scale mixtures of rank-k-continuous axis-aligned p-generalized elliptically contoured Gaussian distributions and answer the question when an n-dimensional rank-k-continuous axis-aligned p-generalized elliptically contoured distribution is representable as a scale mixture of n-dimensional rank-k-continuous p-generalized Gaussian distribution for a suitable mixture distribution of a positive random variable. Based on this class of multivariate probability distributions, we introduce scale mixed p-generalized Gaussian processes having axis-aligned finite dimensional distributions being p-generalizations of elliptical random processes. Additionally, some of their characteristic properties are discussed and approximates of trajectories of several examples such as p-generalized Student-t and p-generalized Slash processes having axis-aligned finite dimensional distributions are simulated with the help of algorithms to simulate rank-k-continuous axis-aligned p-generalized elliptically contoured distributions.
Highlights
Random processes may be constructed and characterized in different ways
For the special case of scale mixed p-generalized Gaussian processes having axis-aligned fdds, basic properties such as characteristic representations, stationary properties and specific closedness properties are studied and certain approximates of their trajectories are simulated. Preparing for these results, we prove firstly that an apec distribution can be represented by a scale mixture of the apec Gaussian distribution if and only if its density-like generator composed with the pth root function, is completely monotone and secondly that the corresponding mixture distribution is in a well defined way closely connected to the inverse Laplace-Stieltjes transform of this composition
While each finite dimensional distribution of an elliptical process is elliptically contoured, the existence of random processes will be shown whose families of fdds consist of apec distributions
Summary
Random processes may be constructed and characterized in different ways. Apart from constructions via families of random variables whose members satisfy, e.g., specific autoregressive relations or are coefficients of specific series representations, the existence of random processes can be studied following the fundamental existence theorem due to Kolmogorov (1933). For the special case of scale mixed p-generalized Gaussian processes having axis-aligned fdds, basic properties such as characteristic representations, stationary properties and specific closedness properties are studied and certain approximates of their trajectories are simulated Preparing for these results, we prove firstly that an apec distribution can be represented by a scale mixture of the apec Gaussian distribution if and only if its density-like generator composed with the pth root function, is completely monotone and secondly that the corresponding mixture distribution is in a well defined way closely connected to the inverse Laplace-Stieltjes transform of this composition. Is called an n-dimensional rank-k-continuous axis-aligned p-generalized elliptically contoured (kapec) distribution with location parameter μ, scaling matrix D and dg g(k,p) and is denoted by AECn,p μ, D, g(k,p). While each finite dimensional distribution (fdd) of an elliptical process is elliptically contoured, the existence of random processes will be shown whose families of fdds consist of apec distributions
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