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On pgeneralized elliptical random processes
Journal of Statistical Distributions and Applications volume 6, Article number: 1 (2019)
Abstract
We introduce rankkcontinuous axisaligned pgeneralized elliptically contoured distributions and study their properties such as stochastic representations, moments, and densitylike representations. Applying the Kolmogorov existence theorem, we prove the existence of random processes having axisaligned pgeneralized elliptically contoured finite dimensional distributions with arbitrary location and scale functions and a consistent sequence of density generators of pgeneralized spherical invariant distributions. Particularly, we consider scale mixtures of rankkcontinuous axisaligned pgeneralized elliptically contoured Gaussian distributions and answer the question when an ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distribution is representable as a scale mixture of ndimensional rankkcontinuous pgeneralized Gaussian distribution for a suitable mixture distribution of a positive random variable. Based on this class of multivariate probability distributions, we introduce scale mixed pgeneralized Gaussian processes having axisaligned finite dimensional distributions being pgeneralizations of elliptical random processes. Additionally, some of their characteristic properties are discussed and approximates of trajectories of several examples such as pgeneralized Studentt and pgeneralized Slash processes having axisaligned finite dimensional distributions are simulated with the help of algorithms to simulate rankkcontinuous axisaligned pgeneralized elliptically contoured distributions.
Introduction
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). The explicit knowledge of the family of finite dimensional distributions (fdds) can be used then to establish some of the properties of the random process by proving corresponding ones of the fdds. Basic technical problems to be solved this way belong to multivariate distribution theory. In the present paper, Kolmogorov’s theorem is used to prove the existence of real valued random processes having axisaligned pgeneralized elliptically contoured (apec) fdds, thus being pgeneralizations of elliptical random processes having axisaligned fdds.
Well studied examples of random processes which can be constructed via Kolmogorov’s existence theorem are real valued Gaussian processes with emphasis on the Brownian motion, see Shiryaev (1996) and Schilling and Partzsch (2014). Apart from further examples as random processes with independent values, random processes with independent increments as well as Markov processes, spherically invariant random processes being also known as elliptical random processes can be constructed this way. The latter are introduced in Vershik (1964) as random processes consisting of quadratically integrable random variables such that if two of them have the same variance, they follow the same distribution. Corresponding characteristic functions and densities are determined in Blake and Thomas (1968). Yao (1973) and Kano (1994) characterize spherically invariant random processes by establishing that their families of fdds are what is called now scale mixtures of Gaussian distributions having one and the same mixture distribution. The notion of a scale mixture but is first introduced in Andrews and Mallows (1974) and, independently, Wise and jun Gallagher (1978) show that an elliptical random process can be represented as a product of a Gaussian process and a positive random variable being independent of it. Additionally, in Huang and Cambanis (1979), the structure of the space of all second order spherically invariant random processes is studied and used to solve nonlinear estimation problems. Finally, based on the concepts of expansive and semiexpansive sequences of elliptically contoured distributions and apart from analogue representation theorems in Yao (1973) and Kano (1994), a formula to determine the corresponding mixture distribution of the family of fdds of a spherically invariant random process is determined in GómezSánchezManzano et al. (2006).
Besides a thematically assorted summary of several articles on the theory of spherically invariant random processes, numerous applications of these random processes such as modelings of bandlimited speech waveform, of radar clutters, of radio propagation disturbances and of equalization and array processing are dealt with in Yao (2003). Furthermore, the author discusses simulations of trajectories of spherically invariant random processes based on the work in Brehm and Stammler (1987), Conte et al. (1991), and Rangaswamy et al. (1995). More recent applications deal with fading models from spherically invariant random processes in Biglieri et al. (2015) and with MIMO radar target localization and performance evaluation under spherically invariant random process clutter in Zhang et al. (2017).
The notion of a scale mixture of Gaussian distributions is introduced in Andrews and Mallows (1974) as the distribution of the product of a Gaussian variable and an independent positive random variable. A multivariate generalization is given in Lange and Sinsheimer (1993). Using numerous equivalent definitions, scale mixtures of Gaussian distributions are also studied in West (1987), Gneiting (1997), Eltoft et al. (2006), GómezSánchezManzano et al. (2006, 2008), and Hashorva (2012). According to Andrews and Mallows (1974), Lange and Sinsheimer (1993), and GómezSánchezManzano et al. (2006), scale mixtures of Gaussian distributions are special cases of elliptically contoured distributions and an elliptically contoured distribution is a scale mixture of a Gaussian distribution if and only if the composition of its density generator and the square root function is completely monotone. Moreover, examples of scale mixtures of Gaussian distributions are Pearson type VII distributions, power exponential distributions as well as Slash distributions.
Applications of scale mixtures of Gaussian distributions are given in the fields of natural images, insurances and quantitative genetic in Wainwright and Simoncelli (2000), Choy and Chan (2003), and GómezSánchezManzano et al. (2008). More recent applications are Gaussian scale mixture models for robust multivariate linear regression with missing data in AlaLuhtala and Piché (2016), testing homogeneity in a scale mixture of Gaussian distributions in Niu et al. (2016), and adaptive robust regression with continuous Gaussian scale mixture errors in Seo et al. (2017).
For any choice of p>0, introducing the notion of a pgeneralization of a spherically invariant random process means the transition from spherically contoured to l_{n,p}symmetric fdds, the transition from regular elliptically contoured to suitably introduced pgeneralized elliptically contoured distributions and the associated consideration of suitable nonEuclidean instead of Euclidean geometries, respectively. To be more specific, a wellknown example is the ndimensional pgeneralized (spherical) Gaussian distribution being introduced already in Subbotin (1923) and having the probability density function (pdf)
and pgeneralized Weibull, Pearson type II and Pearson type VII distributions are dealt with in Gupta and Song (1997). Additionally, a pgeneralized spherical coordinate transformation, a pgeneralized surface content measure as well as numerous pgeneralized probability distributions and statistics such as pgeneralized versions of the χ^{2}, Student and Fisher distributions are considered in Richter (2007); Richter (2009).
The more general class of continuous l_{n,p}symmetric distributions is studied in ArellanoValle and Richter (2012), Kalke and Richter (2013), Müller and Richter (2016a, b, 2017a, b) as well as several references given there. In the present paper, we introduce a class of multivariate apec distributions containing both regular and singular distributions and covering the classes of continuous l_{n,p}symmetric and common axisaligned elliptically contoured distributions.
For a nonempty index set \(I \subseteq \mathbb {R}\), a Polish space (E,ρ) and a family Q of probability measures on the product spaces \(\left (E^{J},\mathcal {B}^{J}\right)\) for nonempty finite subsets J⊆I and the Borel sigma field \(\mathcal {B}\) on E with respect to ρ, if Q is projective on E, Kolmogorov’s existence theorem states the existence of a random process having time set I and state space E such that its family of fdds is equal to Q. The projectivity of Q on E can be shown by checking the consistency conditions in Kolmogorov (1956). This will be discussed for the particular case \(E=\mathbb {R}\) in “Sketch of proof” section. This way, we prove the existence of real valued random processes having apec fdds. Such random processes are pgeneralizations of elliptical random processes having axisaligned fdds. Moreover, for the special case of scale mixed pgeneralized Gaussian processes having axisaligned 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 densitylike 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 LaplaceStieltjes transform of this composition.
The paper is structured as follows. In “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section, ndimensional apec distributions are introduced as locationscale generalizations of continuous l_{n,p}symmetric distributions, and some of their properties such as stochastic representations, moments and pdflike representations are discussed. Furthermore, the pdfs of bivariate pgeneralized spherical as well as of bivariate apec Gaussian distributions are visualized for several values of p>0. Our main result on the existence of pgeneralizations of elliptical random processes is presented in “Main result” section. A sketch of its proof consisting of four basic steps is given in “Sketch of proof” section, and an approximate simulation of the trajectories of the new random processes is discussed in “Simulation” section. Examples illustrating the developed theory are studied in the fourth section. In “Scale mixtures of apec Gaussian distributions” section, scale mixtures of multivariate apec Gaussian distributions are introduced and some of their characteristic properties such as stochastic representations, moments, specific conditional distributions, and their connections to completely monotone functions are discussed. Random processes whose families of fdds are families of scale mixtures of multivariate apec Gaussian distributions with one and the same mixture distribution as well as some of their basic properties are studied in “Scale mixed pgeneralized Gaussian processes having axisaligned fdds” section. All proofs are given in “Proofs” section. For the sake of a better readability, the proofs of certain results are prepared by proving certain particular cases first. An algorithm to simulate arbitrary apec distributions and another one to particularly simulate scale mixtures of apec Gaussian distributions with an explicitly known mixture distribution are presented in Appendix 7.1. The latter one is used in Appendix 7.2 to simulate approximations of trajectories of pgeneralized Studentt as well as pgeneralized Slash processes having axisaligned fdds. Finally, we remark that all figures presented here are made using the program MATLAB.
The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions
For each p>0 and \(n\in \mathbb {N}\), we denote the pfunctional in \(\mathbb {R}^{n}\) by \(x_{p} = \left (\sum \limits _{i=1}^{n}{x_{i}^{p}} \right)^{\frac {1}{p}}\), \(x = (x_{1},\dots,x_{n})^{\text {\texttt {T}}} \in \mathbb {R}^{n}\), and the l_{n,p}generalized surface content of the l_{n,p}unit sphere \(S_{n,p} = \{ x\in \mathbb {R}^{n}\colon x_{p}=1 \}\) by ω_{n,p},
Furthermore, a function g:[0,∞)→[0,∞) satisfying 0<I_{n}(g)<∞ is called a density generating function of an nvariate distribution where \(I_{n}(g) = \int \limits _{0}^{\infty }{ r^{n1} g(r) \:dr}\). An ndimensional random vector \(X\colon \Omega \to \mathbb {R}^{n}\) on a probability space \((\Omega,\mathfrak {A},P)\) having the pdf \(\frac { g\left (\left x\right _{p}\right) }{ \omega _{n,p}\,I_{n}(g) }\), \(x\in \mathbb {R}^{n}\), is called continuous l_{n,p}symmetrically distributed with density generating function g. A density generating function g of a continuous l_{n,p}symmetric distribution satisfying \(I_{n}(g) = \frac {1}{\omega _{n,p}}\) is called a density generator (dg) and denoted by g^{(n,p)}. The pdf of the continuous l_{n,p}symmetric distribution with dg g^{(n,p)} is g^{(n,p)} (x_{p}), \(x\in \mathbb {R}^{n}\), and the corresponding probability law is denoted by \(\Phi _{g^{(n,p)}}\phantom {\dot {i}\!}\). With a view to the special cases listed below, \(\Phi _{g^{(n,p)}}\phantom {\dot {i}\!}\) may also be called ndimensional continuous pgeneralized spherical distribution with dg g^{(n,p)}.
A wellknown example of the latter type of probability distributions is the ndimensional pgeneralized (spherical) Gaussian distribution \(N_{n,p} = \Phi _{g_{PE}^{(n,p)}}\) where
For visualizations of the pdf of this distribution for n∈{1,2} and several p>0, we refer to Kalke and Richter (2013) and Müller and Richter (2015). The class of continuous l_{n,2}symmetric distributions coincides with the class of nvariate continuous spherical distributions and N_{n,2} is the ndimensional standard Gaussian distribution. Numerous properties such as stochastic representations, moments, and marginal distributions and several types of dgs are discussed in Gupta and Song (1997), Richter (2009), ArellanoValle and Richter (2012), and Müller and Richter (2016a).
Let \(\mu \in \mathbb {R}^{n}\) be a constant vector and D=diag(d_{1},…,d_{n}) an n×n diagonal matrix having nonnegative diagonal entries and positive rank rk(Σ)=k. Moreover, let I_{1}={i_{1},…,i_{k}}⊆{1,…,n} with I_{1}=k and i_{1}<i_{2}<…<i_{k} be the set of indices such that d_{i}>0 if i∈I_{1} and d_{i}=0 if i∈I_{2}={1,…,n}∖I_{1}. Let \(e_{i}^{(n)}\) denote the ith unit vector in \(\mathbb {R}^{n} 0_{n \times n}\) the n×n zero matrix, \(S_{1} = \text { diag }\left (d_{i_{1}},\ldots,d_{i_{k}}\right) \in \mathbb {R}^{k \times k}\), \(W_{1} = \left (e_{i_{1}}^{(n)} \cdots \, e_{i_{k}}^{(n)} \right) \in \mathbb {R}^{n \times k}\) and \(W_{2} \in \mathbb {R}^{n \times (nk)}\) a matrix having columns \(e_{i}^{(n)}\) for all i∈I_{2}, then,
Let \(\sqrt {S_{1}} = \text { diag }\left (\sqrt {d_{i_{1}}},\ldots,\sqrt {d_{i_{k}}}\right)\). The distribution of a random vector X satisfying the stochastic representation
is called an ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured (kapec) distribution with location parameter μ, scaling matrix D and dg g^{(k,p)} and is denoted by AEC_{n,p} (μ,D,g^{(k,p)}). For simplicity, the distribution of such random vector X is just called apec distribution if its continuity and dimension as well as the rank of the diagonal matrix parameter D are contextually clear or play only a minor role.
Here and in what follows, X=dZ and X∼Ψ mean that the random vectors X and Z follow the same distribution law and that the random vector X follows the distribution law \(\mathfrak {L}({X}) = \Psi \), respectively. In particular, for the special choice of μ and D to be the zero vector 0_{n} and identity matrix I_{n×n} in \(\mathbb {R}^{n}\), respectively, we have \(AEC_{n,p}\!\left (0_{n},I_{n \times n},g^{(n,p)}\right) = \Phi _{g^{(n,p)}}\). For the special case of p=2, the class of AEC_{n,2} (μ,D,g^{(k,2)})distributions is identical with the class of common nvariate axisaligned elliptically contoured distributions. Furthermore, \(AEC_{n,p}\!\left (\mu,D,g_{PE}^{(k,p)}\right)\) is called ndimensional kapec Gaussian distribution and is denoted AN_{n,p}(μ,D). The family of apec distributions with full rank scaling matrices as well as their starshaped extensions and certain aspects of their inferential applications are studied in Richter (2014, 2016, 2017).
Because of relation (1), a stochastic representation and properties of moments of ndimensional kapec distributions stated in Lemmata 2.1 and 2.2 follow immediately from corresponding results of l_{k,p}symmetric distributions in Richter (2009) and ArellanoValle and Richter (2012).
Lemma 2.1
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where rk(D)=k. Then, the random vector X satisf ies the stochastic representation
where the random vector \(U_{p}^{(k)}\) is kdimensional pgeneralized uniformly distributed on S_{k,p}, R and \(U_{p}^{(k)}\) are stochastically independent and R is a nonnegative random variable with pdf
Lemma 2.2
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where rk(D)=k. Then, \(\mathbb {E}(X) = \mu \) if I_{k+1} (g^{(k,p)}) is f inite and \(\mathrm {Cov(X)} = \sigma _{g^{(k,p)}}^{2} D\) if I_{k+2} (g^{(k,p)}) is f inite where the univariate variance component \(\sigma _{g^{(k,p)}}^{2}\phantom {\dot {i}\!}\) of \(\Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\) satisf ies \(\sigma _{g^{(k,p)}}^{2} = \frac {\Gamma \!\left (\frac {3}{p}\right)\Gamma \!\left (\frac {k}{p}\right)} {\Gamma \!\left (\frac {1}{p}\right)\Gamma \!\left (\frac {k+2}{p}\right)} \, \omega _{k,p} \, I_{k+2}(g^{(k,p)})\). The components of X are independent if and only if \(g^{(k,p)} = g_{PE}^{(k,p)}\).
The justification for calling \(\sigma _{g^{(n,p)}}^{2}\phantom {\dot {i}\!}\) the univariate variance component of \(\Phi _{g^{(n,p)}}\phantom {\dot {i}\!}\) is given by the following lemma with k=1. Examples of \(\sigma _{g^{(n,p)}}^{2}\phantom {\dot {i}\!}\) are given in Müller and Richter (2016b). Let us remark that, according to ArellanoValle and Richter (2012), for k=1,…,n−1, the marginal dg \(g_{(n)}^{(k,p)}\) of an arbitrary kdimensional marginal distribution of \(\Phi _{g^{(n,p)}}\phantom {\dot {i}\!}\) is
where the variability of the choice of the k marginal variables is established by the permutation invariance of \(\Phi _{g^{(n,p)}}\phantom {\dot {i}\!}\), see Müller and Richter (2016b).
Lemma 2.3
For k=1,…,n−1,
Denoting \(M_{n}^{*} = [0,\pi)^{\times (n2)} \times [0,2\pi)\) and \(M_{n} = [0,\infty) \times M_{n}^{*}\) for n≥2, let the l_{n,p}spherical coordinate transformation \(SPH_{p}^{(n)} \colon M_{n} \to \mathbb {R}^{n}\) be defined as in Richter (2007). Note that \(SPH_{p}^{(n)}\) is bijective a.e. in M_{n} and its inverse mapping as well as its Jacobian are explicitly known. The next lemma combines and states more precisely some earlier results and introduces a second stochastic representation of random vectors following the distribution AEC_{n,p} (μ,D,g^{(k,p)}).
Lemma 2.4
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where rk(D)=k. Then, the random vector X satisf ies the stochastic representation
where the nonnegative random variables R,Ψ_{1},…,Ψ_{k−1} are mutually stochastic independent having pdfs
Here, N_{p}(ψ)=(sin(ψ)^{p}+cos(ψ)^{p})^{1/p} and f_{Z} denotes the pdf of Z.
While the distribution AEC_{n,p} (μ,D,g^{(n,p)}) is regular and has a pdf, the distribution AEC_{n,p} (μ,D,g^{(k,p)}) is singular if rk(D)=k<n and may be characterized by a pdflike representation as it was done in Khatri (1968) and Rao (1973, pp. 527528) in case of singular normal distributions and in ArellanoValle and Azzalini (2006, Appendix C) in case of singular unified skewnormal distributions. To this end, let \(U_{W_{2}^{\text {\texttt {T}}}}(\mu) = \{ x\in \mathbb {R}^{n} \colon W_{2}^{\text {\texttt {T}}} x = W_{2}^{\text {\texttt {T}}} \mu \}\) be a kdimensional affine subspace in \(\mathbb {R}^{n}\) and \(\lambda _{U_{W_{2}^{\text {\texttt {T}}}}(\mu)}^{(k)}\) the kdimensional Lebesgue measure defined on \(U_{W_{2}^{\text {\texttt {T}}}}(\mu)\).
Lemma 2.5
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where rk(D)=k. Then, the distribution of X has pdflike representation
where the function given in (2) is interpreted as pdf in the space \(U_{V_{2}^{\text {\texttt {T}}}}(\mu)\) in which the whole probability mass of X is concentrated according to Eq. 3.
Lemma 2.5 can be read as follows. For X∼AEC_{n,p} (μ,D,g^{(k,p)}), the orthogonal projection \(Y = \Pi _{U_{W_{2}^{\text {\texttt {T}}}}(\mu)}(X)\) of X into the subspace \(U_{W_{2}^{\text {\texttt {T}}}}(\mu)\), and any event \(B\in \mathfrak {B}^{n}\),
meaning that the probability measure induced by the random vector X, P^{X}=AEC_{n,p} (μ,D,g^{(k,p)}), is absolutely continuous with respect to \(\lambda _{U_{V_{2}^{\text {\texttt {T}}}}(\mu)}^{(k)}\). Thus, (2) is the RadonNikodym derivative of P^{X} with respect to the Lebesgue measure \(\lambda _{U_{V_{2}^{\text {\texttt {T}}}}(\mu)}^{(k)}\) on the subspace \(U_{W_{2}^{\text {\texttt {T}}}}(\mu)\) of \(\mathbb {R}^{n}\). Because of (4), g^{(k,p)} might be called densitylike generator of AEC_{n,p} (μ,D,g^{(k,p)}) if k<n. In particular, if rk(D)=n, then W_{1}=I_{n×n} and W_{2} is not defined. Hence, Eq. 3 is not applicable and the function in (2) is the common pdf of the distribution AEC_{n,p} (μ,D,g^{(n,p)}). An example is illustrated in Fig. 1.
At the end of this section, our consideration will be slightly extended in order to cover the case k=rk(D)=0 or, equivalently, D=0_{n×n}. To this end, AEC_{n,p} (μ,0_{n×n},g^{(0,p)}) is defined to be the Dirac distribution at \(\mu \in \mathbb {R}^{n}\) where g^{(0,p)} is just a symbol to maintain previous notations.
While each finite dimensional distribution (fdd) of an elliptical process is elliptically contoured, in the next section the existence of random processes will be shown whose families of fdds consist of apec distributions.
Generalized elliptical random processes
3.1 Main result
In order to state our main result, we call a sequence \(g^{(p)} = \left (g^{(k,p)}\right)_{k\in \mathbb {N}}\) of dgs of continuous l_{k,p}symmetric distributions consistent if the following condition is satisfied for any \(k\in \mathbb {N}\) and almost all \(\left (x_{1},\ldots,x_{k}\right)^{\text {\texttt {T}}} \in \mathbb {R}^{k}\),
For the particular case of this definition if p=2, we refer to Kano (1994). Moreover, for any nonempty subset I of \(\mathbb {R}\), any functions \(m \colon I \to \mathbb {R}\) and S:I→[0,∞), and any sequence \(g^{(p)} = \left (g^{(k,p)}\right)_{k\in \mathbb {N}}\) of dgs of continuous l_{k,p}symmetric distributions, let the family
of apec distributions having dgs from g^{(p)} and location and scale functions m and S, respectively, be denoted by \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\). Note that strict positivity of S yields a family \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) containing only regular distributions. In difference to this, allowing S to be nonnegative, the family \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) consists both of regular and singular distributions. In particular, the univariate member of this family corresponding to t∈I such that S(t)=0 is AEC_{1,p} (m(t),0,g^{(0,p)}), i.e. an univariate kapec distribution with k=0.
Theorem 4.1
If g^{(p)} is consistent, then \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) is projective on \(\mathbb {R}\).
Corollary 3.1
According to the Kolmogorov existence theorem, for any nonempty subset I of \(\mathbb {R}\), functions \(m \colon I \to \mathbb {R}\) and S:I→[0,∞), and consistent sequence g^{(p)}, Theorem 4.1 yields the existence of a realvalued random process having \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) as its family of fdds.
A random process defined according to Theorem 4.1 and Corollary 3.1 is called random process having apec fdds with location and scale functions m and S, respectively, and sequence g^{(p)} of dgs of continuous l_{k,p}symmetric distributions. Such random process is denoted by \(\mathop {AEC\!P}_{p}\!\left (m,S;g^{(p)}\right)\).
3.2 Sketch of proof
Because of the complexity of the proof of Theorem 4.1, we first give a sketch of its principal ideas. For the outline of details of proof, we refer to “Proof of Theorem 4.1” section. The first step and fundamental argument to prove Theorem 4.1 and thus the existence of the random processes according to Corollary 3.1 is to show that the family \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) satisfies Kolmogorov’s consistency conditions. Let the set of all finite and nonempty subsets of I be denoted by \(\mathcal {H}(I)\), \(\mathcal {H}(I) = \left \{ J \subseteq I \colon J\neq \emptyset,\, \left J\right <\infty \right \}\). According to Kolmogorov (1956), a family \(Q = \left \{ Q_{J} \right \}_{\{J\in \mathcal {H}(I)\}}\) of probability measures on \(\left (\mathbb {R}^{J},\mathfrak {B}^{J}\right)\), \(J\in \mathcal {H}(I)\), is projective on \(\mathbb {R}\) if the following two conditions are satisfied:

1)
For all t_{1},…,t_{n},t_{n+1}∈I being pairwise distinct and \(A^{(n)} \in \mathfrak {B}^{n}\),
$$ Q_{\{ t_{1},\ldots,t_{n},t_{n+1} \}}\!\left(A^{(n)} \times E \right) = Q_{\{ t_{1},\ldots,t_{n} \}}\!\left(A^{(n)} \right). $$(6) 
2)
For all t_{1},…,t_{n}∈I, \(A^{(n)} \in \mathfrak {B}^{n}\) being pairwise distinct and every permutation π of {1,…,n},
$$ Q_{\{ t_{1},\ldots,t_{n} \}}\!\left(A^{(n)} \right) = Q_{\{ t_{\pi(1)},\ldots,t_{\pi(n)} \}}\!\left(A_{\pi}^{(n)} \right) $$(7)where \(A_{\pi }^{(n)} = \left \{ (x_{\pi (1)},\ldots,x_{\pi (n)})^{\text {\texttt {T}}} \colon \left (x_{1},\ldots,x_{n}\right)^{\text {\texttt {T}}} \in A^{(n)} \right \}\).
These two conditions are traditionally formulated using the notion of ordered sets which are assumed to have different elements, i.e. the sets {t_{1},t_{2}} and {t_{2},t_{1}} differ from each other if t_{1}≠t_{2}, whereas (7) is not required in case of considering unordered sets, see Shiryaev (1996, p. 168).
Condition (6) ensures that specific marginal distributions of elements of the family Q are elements of this family, too. Proving (6) for the family given in Theorem 4.1 will be done in steps two and three. Since both of them are connected with transitions from joint to marginal distributions, we will use the notion of marginal dgs \(g_{(k)}^{(m,p)}\), m=1,…,k−1, according to “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section. Additionally, let \(g_{(k)}^{(k,p)} = g^{(k,p)}\). Making use of the marginal dg, in step two an equivalent formulation of (5) is given in the next lemma.
Lemma 3.1
A sequence \(g^{(p)} = \left (g^{(k,p)}\right)_{k\in \mathbb {N}}\) of dgs of continuous l_{k,p}symmetric distributions is consistent if and only if for any \(k\in \mathbb {N}\)
As a consequence, a sequence g^{(p)} of dgs of continuous l_{k,p}symmetric distributions is consistent if and only if for any \(k\in \mathbb {N}\) the marginal dg \(g_{(k+1)}^{(k,p)}\) corresponding to the (k+1)th element g^{(k+1,p)} of g^{(p)} coincides with the kth element g^{(k,p)}. In the third step, for m≤n, mdimensional marginal distributions of ndimensional apec distributions are shown to be mdimensional apec distributions with suitably modified vector and matrix parameters and transitions to marginal dgs.
Lemma 3.2
For \(\mu = \left (\mu _{1},\ldots,\mu _{n}\right)^{\text {\texttt {T}}}\in \mathbb {R}^{n}\) and D= diag (d_{1},…,d_{n}) having nonnegative diagonal entries and rank k≥0, let X=(X_{1},…,X_{n})^{T}∼AEC_{n,p} (μ,D,g^{(k,p)}). Further, let \(m\in \mathbb {N}\) with m≤n, J={j_{1},…,j_{m}}⊆{1,…,n} with j_{1}<…<j_{m}, and \(X_{J} = \left (X_{j_{1}},\dots,X_{j_{m}} \right)^{\text {\texttt {T}}}\) the corresponding mdimensional subvector of X. Then,
where \(\mu _{J} = \left (\mu _{j_{1}},\ldots,\mu _{j_{m}}\right)^{\text {\texttt {T}}}\), \(D_{J} = \text { diag }\!\left (d_{j_{1}},\ldots,d_{j_{m}} \right)\), and k_{J}=rk(D_{J})≥0.
In the final step four, condition (7) ensures that the considered family of probability distributions is big enough in a suitable sense. Its proof in case of \(Q = \mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) is based on the next lemma on distributions of specific linear transformations of random vectors following an apec distribution.
Lemma 3.3
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) with rk(D)=k≥0. Then, for every (n×n)permutation matrix M and every \(b \in \mathbb {R}^{n}\),
These sketched four steps to prove Theorem 4.1 are outlined in detail in “Proof of Theorem 4.1” section in reverse order. At the end of the present section, we consider an example of random processes being defined by Theorem 4.1 and Corollary 3.1. More general examples are studied in “Scale mixtures and particular pgeneralizations of elliptical random processes” section.
Example 3.1
Let \(g_{PE}^{(p)} = \left (g_{PE}^{(k,p)}\right)_{k\in \mathbb {N}}\) be the sequence of all dgs of multivariate pgeneralized Gaussian distributions. Then, the consistency of \(g_{PE}^{(p)}\) is immediately seen and for any nonempty subset I of \(\mathbb {R}\) and any functions \(m \colon I \to \mathbb {R}\) and S:I→[0,∞), Theorem 4.1 yields the existence of the realvalued random process AGP_{p}(m,S) having \(\mathcal {AEC}_{g_{PE}^{(p)}}^{I}(m,S)\) as its family of fdds. Such stochastic process is called pgeneralized Gaussian process having axisaligned fdds.
3.3 Simulation
In order to simulate a random process X having apec fdds, we consider I=[0,1], simulate the marginal vector of X regarding to the equidistant partition \(\left \{ \frac {i}{200} \colon i=0,\ldots,200 \right \}\) of [0,1] to get a realization of the random vector \(\left (X_{0},X_{\frac {1}{200}},\ldots,X_{\frac {199}{200}},X_{1}\right)^{\text {\texttt {T}}}\). Then, we connect the components of this realization in ascending order by linear functions to get an approximate realization of a trajectory of X. Since components of apec Gaussian distributed random vectors are independent, simulation of the random process AGP_{p}(m,S) according to the method described above is just the simulation of 201 univariate pgeneralized Gaussian variables having specific location and scale parameters. We denote functions on [0,1] taking constant values 0 and 1 by 0_{[0,1]} and 1_{[0,1]}, respectively. Results of the simulation of the random process AGP_{p} (0_{[0,1]},1_{[0,1]}) are shown for \(p\in \left \{\frac {1}{2},1,2,3\right \}\) in Fig. 2. Note that scales of axes are highly dependent on the value of p, but also on the specific realization of a trajectory of the process. Moreover, in Fig. 3, the effect different location and scale functions m and S have on simulations of AGP_{3}(m,S) are shown. See also Appendix 7.2 for several other simulations of random processes having apec fdds.
Scale mixtures and particular pgeneralizations of elliptical random processes
4.1 Scale mixtures of apec Gaussian distributions
Let be \(\mu \in \mathbb {R}^{n}\), \(D\in \mathbb {R}^{n \times n}\) a diagonal matrix having nonnegative diagonal elements and rank k≥0, V a positive random variable, and Z∼AN_{n,p}(0_{n},D) independent of V. Furthermore, let G denote the cumulative distribution function (cdf) of V. Then, the distribution of an ndimensional random vector X satisfying the stochastic representation
is called scale mixture of the ndimensional kapec Gaussian distribution with parameters μ and D and with mixture cdf G and is denoted by SMAN_{n,p}(μ,D,G).
The particular cases SMAN_{1,2}(0,1,G), SMAN_{n,2}(μ,D,G) with full rank matrix D, and SMN_{n,p}(G)=SMAN_{n,p}(0_{n},I_{n×n},G) are introduced in Andrews and Mallows (1974), Lange and Sinsheimer (1993), and ArellanoValle and Richter (2012), respectively, where numerous equivalent parameterizations of scale mixtures of the common multivariate Gaussian distribution and different notions such as normal/independent distributions or variance mixtures of Gaussian distribution are used. As a first characterization of the class of SMAN_{n,p}(μ,D,G)distributions, its connections to the classes of SMN_{n,p}(G) and AEC_{n,p}(μ,D,g^{(k,p)})distributions are studied next.
Lemma 4.1
A random vector \(X\colon \Omega \to \mathbb {R}^{n}\) satisf ies X∼SMAN_{n,p}(μ,D,G)with k=rk(D)≥1 if and only if
Corollary 4.1
There holds \(SMAN_{n,p}(\mu,D,G) = AEC_{n,p}\!\left (\mu,D,g_{SMN;G}^{(k,p)}\right)\) with k=rk(D) and
As a result, scale mixtures of kapec Gaussian distributions are themselves kapec. Moreover, many properties of such scale mixtures (such as stochastic representations according to Lemmata 2.1 and 2.4) can be obtained from properties of ndimensional kapec distributions by specializing dgs (according to that given in Corollary 4.1). Additionally, some properties as the first two moments of SMAN_{n,p}(μ,D,G) can be specialized as follows.
Corollary 4.2
Let X∼SMAN_{n,p}(μ,D,G) with k=rk(D)≥1 and V∼G. Then, \(\mathbb {E}(X)=\mu \) if \(\mathbb {E}\left (V^{\frac {1}{p}}\right)\) is f inite, and \(\mathrm {Cov(X)} = \sigma _{g_{SMN;G}^{(k,p)}}^{2} D\) if \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is f inite where
Because of the assertion of the following lemma, SMAN_{n,p}(μ,D,G) can be called a variance mixture of AN_{n,p}(μ,D). In the special case of μ=0_{n}, D=I_{n×n} and p=2, the following lemma is covered by the main theorem in Kingman (1972).
Lemma 4.2
Let X∼SMAN_{n,p}(μ,D,G) with k=rk(D)≥1 and V∼G a positive random variable. Then, the conditional distribution of X given V=v satisf ies
According to Corollary 4.1, each scalemixture of the ndimensional apec Gaussian distribution is an ndimensional apec distribution with a specific dg. Now, we are interested in which AEC_{n,p} (μ,D,g^{(k,p)})distributions can be represented by scale mixtures of the ndimensional apec Gaussian distribution. This question is answered by the following theorem using the notion of completely monotone functions on [0,∞). A function \(f\colon (0,\infty)\to \mathbb {R}\) is called completely monotone if its restriction f^{∗}=f_{(0,∞)} to (0,∞) is completely monotone, i.e. f^{∗} is infinitely often differentiable and satisfies the inequality \(\left (1\right)^{m} \frac {d^{m}f}{dx^{m}}(z) \geq 0\) for all z∈(0,∞) and all \(m\in \mathbb {N}_{0}=\mathbb {N}\cup \{0\}\), see Sasvári (2013).
Theorem 4.1
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) with D having positive rank k. Then, X∼SMAN_{n,p}(μ,D,G) for the cdf G of a suitable positive random variable if and only if the function h def ined by \(h(y) = g^{(k,p)}\!\left (\sqrt [p]{y}\right)\), y∈[0,∞), is completely monotone.
For the special case of n=1 and p=2, this theorem is proven in Andrews and Mallows (1974). Subsequently, the Euclidean case p=2 of Theorem 4.1 in arbitrary dimensions (\(n\in \mathbb {N}\)) is proven in Lange and Sinsheimer (1993) and GómezSánchezManzano et al. (2006). Particularly, the proof of Theorem 4.1 given in “Proofs regarding to “Scale mixtures of apec Gaussian distributions” section“Scale mixtures of apec Gaussian distributions” section has analogies to that in Andrews and Mallows (1974) and the cdf G of the corresponding mixture distribution can be determined with the help of the inverse LaplaceStieltjes transform of h.
Corollary 4.3
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) with k=rk(D)≥1 and assume that the function \(y \mapsto g^{(k,p)}\!\left (\sqrt [p]{y}\right)\) is completely monotone in (0,∞) and has the inverse LaplaceStieltjes transform α, that is
Then, X∼SMAN_{n,p}(μ,D,G) and the cdf G of the mixture distribution satisf ies the representation
Moreover, the probability law corresponding to G is regular and has pdf f_{G} if and only if α is absolutely continuous with pdf f_{α} and both pdfs are connected by the equation
Example 4.1
An ndimensional apec Gaussian distribution is a scale mixture of itself with the Dirac distribution in 1 being the mixture distribution. The cdf of this Dirac distribution is the indicator function \(s \mapsto \mathbbm{1}_{(1,\infty)}(s)\).
Example 4.2
The ndimensional kapec Pearsontype VII distribution with parameters M and ν, \(M > \frac {k}{p}\) and ν>0, and dg
is the scale mixture of the ndimensional kapec Gaussian distribution where the mixture distribution is the Gamma distribution \(\Gamma _{M\frac {k}{p},\frac {\nu }{p}}\) having pdf
Example 4.3
A special case of the preceding one is the ndimensional kapec Studentt distribution with parameter ν>0 and dg \(g_{St;\nu }^{(k,p)} = g_{PT7;\frac {\nu +k}{p},\nu }^{(k,p)}\) being that of the scale mixture of the ndimensional kapec Gaussian distribution with mixture distribution \(\Gamma _{\frac {\nu }{p},\frac {\nu }{p}}\).
Example 4.4
The ndimensional kapec Slash distribution with parameter ν>0 is def ined as the scale mixture of the ndimensional kapec Gaussian distribution with mixture distribution having pdf \(f_{\nu }^{Sl}(y) = \nu y^{\nu 1} \mathbbm{1}_{(0,1)}(y)\), \(y\in \mathbb {R}\).
4.2 Scale mixed pgeneralized Gaussian processes having axisaligned fdds
Let \(g_{SMN;G}^{(p)} = \left (g_{SMN;G}^{(k,p)}\right)_{k\in \mathbb {N}}\) denote the sequence of dgs of scale mixtures of kdimensional pgeneralized Gaussian distributions with one and the same mixture cdf G with
According to Examples 4.14.4, representatives of mixture cdfs satisfying (9) are the Dirac distribution in 1, \(\Gamma _{\frac {\nu }{p},\frac {\nu }{p}}\) as well as the distribution with pdf \(f_{\nu }^{Sl}\), whereas the cdf of the distribution \(\Gamma _{M\frac {k}{p},\frac {\nu }{p}}\) does not generally satisfy (9).
Lemma 4.3
For the cdf G of a positive random variable satisfying (9), the sequence \(g_{SMN;G}^{(p)}\) is consistent.
Throughout this section, again let I be a nonempty subset of \(\mathbb {R}\), \(m \colon I \to \mathbb {R}\) and S:I→[0,∞) arbitrary functions, and G the cdf of a positive random variable satisfying (9). Then, a random process having apec fdds, location and scale functions m and S, respectively, and the sequence \(g_{SMN;G}^{(p)}\) of dgs exists according to Theorem 4.1 and Corollary 3.1. Such process is called a scale mixed pgeneralized Gaussian process having axisaligned fdds with location function m, scale function S and mixture cdf G and is denoted by SMAGP_{p}(m,S,G), thus \(AECP_{p}\!\left (m,S;g_{SMN;G}^{(p)}\right) = SMAGP_{p}(m,S,G)\). The motivation and justification of this naming is given by a characterizing property of such processes in Theorem 4.2 below.
On the one hand, for the special case p=2, the class of SMAGP_{p}(m,S,G)processes is equal to the class of spherically invariant random processes having axisaligned fdds which is defined in Vershik (1964). Moreover, it is shown implicitly in Yao (1973) and explicitly in Kano (1994) that a sequence g^{(2)} is consistent if and only if all elements of g^{(2)} are dgs of scale mixtures of multivariate Gaussian distributions regarding to one and the same mixture distribution. On the other hand, for general p>0, if the mixture distribution is chosen to be the Dirac distribution in 1, then \(SMAGP_{p}\!\left (m,S,\mathbbm{1}_{(1,\infty)}\right) = AGP_{p}(m,S)\). Furthermore, for any ν>0, let us denote the cdf of \(\Gamma _{\frac {\nu }{p},\frac {\nu }{p}}\) and of the distribution with pdf \(f_{\nu }^{Sl}\) by \(G_{\nu /p}^{St}\) and \(G_{\nu }^{Sl}\), respectively. Then, \(SMAGP_{p}\!\left (m,S,G_{\nu /p}^{St}\right)\) and \(SMAGP_{p}\!\left (m,S,G_{\nu }^{Sl}\right)\) are called pgeneralized Studentt and pgeneralized Slash process having axisaligned fdds with location function m, scale function S and parameter ν, and are denoted by AStP_{p}(m,S,ν) and ASlP_{p}(m,S,ν), respectively.
Because of its construction, a scale mixed pgeneralized Gaussian process X having axisaligned fdds with location function m, scale function S and mixture cdf G is uniquely determined except for equivalence and denoted X∼SMAGP_{p}(m,S,G). Next, we state a characteristic representation of the random process SMAGP_{p}(m,S,G) with the help of a specific pgeneralized Gaussian process providing the motivation for the naming of such process.
Theorem 4.2
Let X={X_{t}}_{t∈I} be a scale mixed pgeneralized Gaussian process having axisaligned fdds, X∼SMAGP_{p}(m,S,G). Then, X and \(Y = \left \{ m(t) + V^{\frac {1}{p}} Z_{t} \right \}_{t \in I}\) are equivalent where the pgeneralized Gaussian process Z={Z_{t}}_{t∈I}∼AGP_{p}(0_{I},S) having axisaligned fdds is independent of the random variable V∼G.
For p=2 and m=0_{I}, Theorem 4.2 is proven in Wise and jun Gallagher (1978). In the sequel, using the characteristic representation from Theorem 4.2, we determine expectation and covariance functions as well as stationarity properties of the random process SMAGP_{p}(m,S,G). Since SMAGP_{p}(m,0_{I},G) equals a.s. the location function m, the results of Theorems 4.3 and 4.5 below are restricted to nonvanishing scale functions, i.e. S≠0_{I}. Let \(g_{SMN;G}^{(p)} = \left (g_{SMN;G}^{(k,p)}\right)_{k\in \mathbb {N}}\) be the sequence of dgs of scale mixtures of multivariate pgeneralized Gaussian distributions with one and the same mixture cdf G such that \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is finite where V∼G. Then, because of Corollary 4.2 and property (9) of G, the sequence \(\left (\sigma _{g_{SMN;G}^{(k,p)}}^{2} \right)_{k\in \mathbb {N}}\) of the corresponding univariate variance components is constant and an arbitrary element of it is subsequently denoted by \(\sigma _{g_{SMN;G}^{(p)}}^{2}\phantom {\dot {i}\!}\).
Theorem 4.3
Let X={X_{t}}_{t∈I}∼SMAGP_{p}(m,S,G) with S≠0_{I} and V∼G. Then, the expectation function of the random process X exists and is equal to the location function m if \(\mathbb {E}\left (V^{\frac {1}{p}}\right)\) is f inite. If \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is f inite, X is a second order random process with covariance function \(\Gamma \colon I \times I \to \mathbb {R}\) given by
As announced before, different stationarity properties of the random process SMAGP_{p}(m,S,G) are studied now. We start with a result on strict stationarity.
Theorem 4.4
Let X={X_{t}}_{t∈I}∼SMAGP_{p}(m,S,G). Then, X is strictly stationary if and only if m and S are constant.
In the following theorem, we additionally take the notions of weak stationarity and white noise into consideration.
Theorem 4.5
Let X={X_{t}}_{t∈I}∼SMAGP_{p}(m,S,G), V∼G, \(\mu \in \mathbb {R}\) and δ>0. Then, the following statements are equivalent:

1)
There holds m(t)=μ and S(t)=δ for all t∈I and \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is finite.

2)
X is strictly stationary, \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is finite, the expectation function of X attains the constant value μ and the covariance function Γ of X satisfies \(\Gamma (t,t) = \sigma _{g_{SMN;G}^{(p)}}^{2} \delta \) for all t∈I and Γ(s,t)=0 for all s,t∈I with s≠t.

3)
X is weakly stationary with constant expectation μ and covariance function Γ given by Γ(s,t)=K(s−t) where K satisfies \(K(0) = \sigma _{g_{SMN;G}^{(p)}}^{2} \delta \) and K(h)=0 for all h∈{s−t:s,t∈I}∖{0}.

4)
X is white noise with expectation μ and variance \(\sigma _{g_{SMN;G}^{(p)}}^{2} \delta \).
Finally, we establish the closedness of the class of all scale mixed pgeneralized Gaussian processes having axisaligned fdds with respect to linear transformations.
Theorem 4.6
Let {X_{t}}_{t∈I}∼SMAGP_{p}(m,S,G), \(b \colon I \to \mathbb {R}\) and \(\gamma \colon I \to \mathbb {R}\). Then,
where \(\lbrack \gamma m + b\rbrack \colon I \to \mathbb {R}\) and [γ^{2}S]:I→[0,∞) are def ined by [γm+b](t)=γ(t)m(t)+b(t), t∈I, and [γ^{2}S](t)=(γ(t))^{2}S(t), t∈I, respectively.
Proofs
5.1 Proofs of Lemmata 2.3 and 2.5
Before proving Lemma 2.3, we state a part of its proof as the following remark on the pgeneralized surface content of pgeneralized spheres of different dimensions in relation with a certain integral.
Remark 5.1
For every \(\nu \in \mathbb {N}\) with ν≥2 and every κ∈{1,…,ν−1},
According to Richter (2009), the left hand side of the above equation is the limit of the cdf of the pgeneralized Fisher statistic T_{ν−κ,κ}(p) evaluated at t as t→∞. Hence, Remark 5.1 follows from the elementary fact that the cdf of a univariate random variable evaluated at t tends to one as t→∞.
Proof of Lemma 2.3
Let k∈{1,…,n−1} be fixed. Denoting \(\tau _{k,p} = \frac { \Gamma \!\left (\frac {3}{p}\right)\Gamma \!\left (\frac {k}{p}\right) }{\Gamma \!\left (\frac {1}{p}\right)\Gamma \!\left (\frac {k+2}{p}\right) }\), using integral transformation y=z^{p}+r^{p} with \(\frac {dy}{dz} = pz^{p1}\) and finally renaming r and z by x and y, respectively, we get
Applying the l_{2,p}spherical coordinate transformation \(x = r \frac {\cos (\psi)}{N_{p}(\psi)}\) and \(y = r \frac {\sin (\psi)}{N_{p}(\psi)}\) with N_{p}(ψ)=(sin(ψ)^{p}+cos(ψ)^{p})^{1/p} and \(\frac {d(x,y)}{d(r,\psi)} = \frac {r}{N_{p}^{2}(\psi)}\), see Richter (2007), Fubini’s theorem and Remark 5.1 for ν=n+2 and κ=n−k, it follows
□
Proof of Lemma 2.5
It follows from \(D = \left (W_{1}\sqrt {S_{1}}\right)\!\left (W_{1}\sqrt {S_{1}}\right)^{\text {\texttt {T}}}\) and Lemma 2.1 that
Since \(\sqrt {S_{1}}\) has full rank k, \(W_{1}^{\text {\texttt {T}}} X\) is kdimensional rankkcontinuous pgeneralized elliptically contoured distributed with parameters \(W_{1}^{\text {\texttt {T}}} \mu \) and S_{1} and with dg g^{(k,p)}. By definition of this distribution, for \(\phantom {\dot {i}\!}Y \sim \Phi _{g^{(k,p)}}\), it follows \(W_{1}^{\text {\texttt {T}}} X \stackrel {d}{=} W_{1}^{\text {\texttt {T}}} \mu + \sqrt {S_{1}} Y\phantom {\dot {i}\!}\). Thus, \(\phantom {\dot {i}\!}W_{1}^{\text {\texttt {T}}} X\) has pdf
Since the columns of W_{1} and W_{2} together build an orthonormal basis of \(\mathbb {R}^{n}\), we have \(W_{2}^{\text {\texttt {T}}} W_{1} = 0_{(nk) \times k}\phantom {\dot {i}\!}\) and
Thus, the orthogonal projection \(\phantom {\dot {i}\!}Y = \Pi _{U_{W_{2}^{\text {\texttt {T}}}}(\mu)}(X)\) of X into the space \(U_{W_{2}^{\text {\texttt {T}}}}(\mu)\phantom {\dot {i}\!}\) has the pdf
and the orthogonal projection of X into the orthogonal complement of \(U_{W_{2}^{\text {\texttt {T}}}}(\mu)\phantom {\dot {i}\!}\) has probability mass zero. □
5.2 Proof of Theorem 4.1
We start with considering a particular case of Lemma 3.3.
Lemma 5.1
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where rk(D)=k≥1. Then, for every (n×n)permutation matrix M and every \(b \in \mathbb {R}^{n}\),
Proof
With notations from “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section,
Since \(\phantom {\dot {i}\!}M W_{1} \sqrt {S_{1}} \in \mathbb {R}^{n \times k}\) arises from \(W_{1} \sqrt {S_{1}}\) by interchanging rows, it has rank k. Thus,
□
Proof of Lemma 3.3
Because of Lemma 5.1, only the case k=0 has to be considered. In this case, X∼AEC_{n,p} (μ,0_{n×n},g^{(0,p)}), i.e. X follows the Dirac distribution in μ. Thus,
and \(\mathfrak {L}(MX+b) = AEC_{n,p}\!\left (M\mu +b,M 0_{n \times n} M^{\text {\texttt {T}}},g^{(0,p)}\right)\) because of 0_{n×n}=M0_{n×n}M^{T}. □
Denoting the cardinality of the set A by A, we continue with studying a particular case of Lemma 3.2.
Lemma 5.2
Let be X=(X_{1},…,X_{n})^{T}∼AEC_{n,p} (μ,D,g^{(k,p)}) where \(\mu = \left (\mu _{1},\ldots,\mu _{n}\right)^{\text {\texttt {T}}}\in \mathbb {R}^{n}\) and assume D= diag (d_{1},…,d_{n}) has nonnegative diagonal entries and rank k≥1. Further, let \(m\in \mathbb {N}\) with m≤n, J={j_{1},…,j_{m}}⊆{1,…,n} with j_{1}<…<j_{m} and \(\left \left \{ \eta \in \{1,\ldots,m\} \colon d_{j_{\eta }} > 0 \right \}\right  \geq 1\). Then, the corresponding mdimensional subvector \(X_{J} = \left (X_{j_{1}},\dots,X_{j_{m}} \right)^{\text {\texttt {T}}}\) of X satisf ies
where \(\mu _{J} = \left (\mu _{j_{1}},\ldots,\mu _{j_{m}}\right)^{\text {\texttt {T}}}\), \(D_{J} = \text { diag }\!\left (d_{j_{1}},\ldots,d_{j_{m}} \right)\) and k_{J}=rk(D_{J})≥1.
Proof
Starting from the equation X_{J}=ΓX where
and using notations from “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section, it follows that
where
Thus, for \(Y = \left (Y_{1},\ldots,Y_{k}\right) \sim \Phi _{g^{(k,p)}}\), we get
where
η=1,…,m. Now, let
Then, \(\left K\right  = \left \left \{ \eta \in \{1,\ldots,m\} \colon \sigma _{j_{\eta }}^{2} > 0 \right \}\right  \geq 1\) and the matrix \(\Gamma W_{1} \sqrt {S_{1}}\) has k−K zero columns. Since each nonzero column is the product of a positive constant with a unit vector in \(\mathbb {R}^{m}\), the vector \(\Gamma W_{1} \sqrt {S_{1}} Y\) consists of K different components of Y multiplied by positive constants and of m−K zeros. Subsequently, put K={l_{1},…,l_{K}} where \(l_{1} < l_{2} < \ldots < l_{K}\phantom {\dot {i}\!}\) is an increasing enumeration of the elements of K and let
be a matrix consisting of the row vectors
Then, for \(B\in \mathfrak {B}^{m}\), \(Y \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\) and \(Z \sim \Phi _{g_{(k)}^{(K,p)}}\phantom {\dot {i}\!}\), it follows that
and, because of (1) and rk(M)=K,
Note that M can be extended to \(\Gamma W_{1} \sqrt {S_{1}}\) by adding zero columns without changing the rank. Moreover, \(MM^{\text {\texttt {T}}} = \left (\Gamma W_{1} \sqrt {S_{1}}\right)\!\left (\Gamma W_{1} \sqrt {S_{1}}\right)^{\text {\texttt {T}}} = \Gamma D \Gamma = D_{J}\) and K=rk(M)=rk(MM^{T})=rk(D_{J})=k_{J}. Summarizing all, we have
□
Proof of Lemma 3.2
If k=0, X∼AEC_{n,p} (μ,0_{n×n},g^{(0,p)}) and J={j_{1},…,j_{m}}⊆{1,…,n} with j_{1}<…<j_{m}. In this case, X_{J}=μ_{J}Pa.s. and
because the symbols g^{(0,p)} and \(g_{(k)}^{(0,p)}\) can be switched for a \(k\in \mathbb {N}\cup \{0\}\). Now, let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where D= diag (d_{1},…,d_{n}) has nonnegative diagonal elements and positive rank k, and let J={j_{1},…,j_{m}}⊆{1,…,n} be an index set such that j_{1}<…<j_{m} and \(\left \left \{ \eta \in \{1,\ldots,m\} \colon \sqrt {d_{j_{\eta }}} > 0 \right \}\right  \geq 1\). Then, Lemma 5.2 yields the assertion. Finally, let X∼AEC_{n,p} (μ,D,g^{(k,p)}) where D= diag (d_{1},…,d_{n}) has nonnegative diagonal elements and positive rank k but, now, where J={j_{1},…,j_{m}}⊆{1,…,n} is an index set such that j_{1}<…<j_{m} and \(\left \left \{ \eta \in \{1,\ldots,m\} \colon \sqrt {d_{j_{\eta }}} > 0 \right \}\right  = 0\). Using the notation from the proof of Lemma 5.2, the set K defined in (10) has cardinality \(\left K\right  = \left \left \{ \eta \in \{1,\ldots,m\} \colon \sqrt {d_{j_{\eta }}} > 0 \right \}\right  = 0\). Because of this, \(\Gamma W_{1} \sqrt {S_{1}}\) is equal to the (m×k) zero matrix and the distribution of \(\Gamma W_{1} \sqrt {S_{1}} Y\) for \(\phantom {\dot {i}\!}Y \sim \Phi _{g^{(k,p)}}\) is concentrated in 0_{m}. Since \(D_{J} = \text { diag }\!\left (d_{j_{1}},\ldots,d_{j_{m}} \right) = 0_{m \times m}\), k_{J}=rk(D_{J})=0 and \(X_{J} = \Gamma X \stackrel {d}{=} \mu _{J} + \Gamma W_{1} \sqrt {S_{1}} Y\) for \(Y \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\), it follows
i.e. \(X_{J} \sim AEC_{m,p}\!\left (\mu _{J},0_{m \times m},g_{(k)}^{(0,p)}\right)\). □
Proof of Lemma 3.1
Starting from (5) and using the transformation \(\tilde {y}=\left x\right _{p}^{p}+y^{p}\), for \(x\in \mathbb {R}^{k}\), we get
Because of ω_{1,p}=2,
□
Proof of Theorem 4.1
For \(n\in \mathbb {N}\) and arbitrary elements t_{1},…,t_{n},t_{n+1} of I, let \(\mu ^{(n+1)} = \left (m(t_{1}),\ldots,m(t_{n}),m(t_{n+1}) \right)^{\text {\texttt {T}}} \in \mathbb {R}^{n+1}\) and assume D^{(n+1)}= diag (S(t_{1}),…,S(t_{n}),S(t_{n+1})) to have rank k. Further, let \(Q_{\left \{ t_{1},\ldots,t_{n},t_{n+1} \right \}}(\cdot) = AEC_{n+1,p}\!\left (\cdot \bigm  \mu ^{(n+1)},D^{(n+1)},g^{(k,p)}\right) \in \mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) be the probability measure induced by a random vector following the (n+1)dimensional kapec distribution with parameters μ^{(n+1)} and D^{(n+1)} and dg g^{(k,p)}∈g^{(p)} if k>0 and symbol g^{(0,p)} if k=0, respectively. By Lemma 3.2, it follows
where μ^{(n)}=(m(t_{1}),…,m(t_{n}))^{T} and D^{(n)}= diag (S(t_{1}),…,S(t_{n})) with κ=rk(D^{(n)})∈{k−1,k}. Furthermore, using Lemma 3.1 if κ>0 and recalling the exchangeability of symbols \(g_{(k)}^{(0,p)}\) and g^{(0,p)} (to maintain the notation as in the proof of Lemma 3.2) if κ=0, we have
Therefore, the marginal probability measure \(Q_{\left \{ t_{1},\ldots,t_{n} \right \}}\) of \(Q_{\left \{ t_{1},\ldots,t_{n+1} \right \}}\) corresponds to the element AEC_{n,p} (μ^{(n)},D^{(n)},g^{(κ,p)}) of \(\mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) and, thus, the Kolmogorov consistency condition (6) is satisfied. Now, let π be a permutation of {1,…,n} and M the corresponding permutation matrix. Additionally, let \(Q_{\left \{ t_{1},\ldots,t_{n} \right \}}(\cdot) = AEC_{n,p}\!\left (\cdot \bigm  \mu,D,g^{(\kappa,p)}\right) \in \mathcal {AEC}_{g^{(p)}}^{I}(m,S)\) be the probability measure induced by a random vector X with X∼AEC_{n,p} (μ,D,g^{(κ,p)}) where μ=μ^{(n)}=(m(t_{1}),…,m(t_{n}))^{T} and D=D^{(n)}= diag (S(t_{1}),…,S(t_{n})) with κ=rk(D). Then, \(Q_{\left \{ t_{\pi (1)},\ldots,t_{\pi (n)} \right \}}\) is induced by MX and, according to Lemma 3.3,
If κ=0, then D=MDM^{T}=0_{n×n}. In this case, \(Q_{\left \{ t_{1},\ldots,t_{n} \right \}}\) and \(Q_{\left \{ t_{\pi (1)},\ldots,t_{\pi (n)} \right \}}\) are Dirac measures in μ and Mμ, respectively, and, for \(A\in \mathfrak {B}^{n}\),
Thus, the Kolmogorov consistency condition (7) is satisfied if κ=0. Now, let κ>0. Using the notations of matrices S_{1}, W_{1} and W_{2} from “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section, \(\left (W_{1}\sqrt {S_{1}}\right)\!\left (W_{1}\sqrt {S_{1}}\right)^{\text {\texttt {T}}}\) is a decomposition of D with \(W_{1}\sqrt {S_{1}} \in \mathbb {R}^{n \times \kappa }\) and \(\mathrm {rk\left (W_{1}\sqrt {S_{1}}\right)} = \kappa \) and the columns of W_{2} are a basis of the kernel of D. Consequently, on the one hand, \(\left (M W_{1} \sqrt {S_{1}}\right)\!\left (M W_{1} \sqrt {S_{1}}\right)^{\text {\texttt {T}}}\) is a corresponding decomposition of MDM^{T} with \(M W_{1}\sqrt {S_{1}} \in \mathbb {R}^{n \times \kappa }\) and \(\mathrm {rk\left (M W_{1}\sqrt {S_{1}}\right)} = \kappa \) since left multiplication of \(W_{1}\sqrt {S_{1}}\) by permutation matrix M only interchanges columns and leaves the rank unchanged. On the other hand, the columns of MW_{2} build a basis of the kernel of MDM^{T},
and
where \(f_{M^{\text {\texttt {T}}}}\phantom {\dot {i}\!}\) is defined by \(\phantom {\dot {i}\!}f_{M^{\text {\texttt {T}}}}(x) = M^{\text {\texttt {T}}} x\), \(x\in \mathbb {R}^{n}\). Finally, for \(A\in \mathfrak {B}^{n}\), Eq. 4 resulting from the pdflike representation of an ndimensional apec distribution together with the transformation \(y = f_{M^{\text {\texttt {T}}}}(x)\phantom {\dot {i}\!}\) having the Jacobian det(M)=1 yield
Thus, the Kolmogorov consistency condition (7) is satisfied in case κ>0, too. □
5.3 Proofs regarding to “Scale mixtures of apec Gaussian distributions” section
Proof of Lemma 4.1
For a positive random variable V∼G, because of (1) and (8), we have \(X \stackrel {d}{=} \mu + V^{\frac {1}{p}} \cdot Z\) where Z∼AN_{n,p}(0_{n},Σ). It follows \(X \stackrel {d}{=} \mu + V^{\frac {1}{p}} \cdot W_{1} \sqrt {S_{1}} \tilde {Z}\) where \(\tilde {Z} \sim N_{k,p}\) and \(X \stackrel {d}{=} \mu + W_{1} \sqrt {S_{1}} V^{\frac {1}{p}} \cdot \tilde {Z}\) where \(\tilde {Z} \sim N_{k,p}\). Thus, \(X \stackrel {d}{=} \mu + W_{1} \sqrt {S_{1}} \tilde {X}\) where \(\tilde {X} \sim SMN_{k,p}(G)\). □
Proof of Corollary 4.1
In the case k≥1, the assertion follows from Lemma 4.1, Eq. 1 and the identity \(SMN_{k,p}(G) = \Phi _{g_{SMN;G}^{(k,p)}}\) from ArellanoValle and Richter (2012). In the case k=0, Z=0_{n} a.s. in (8). Therefore, X∼SMAN_{n,p}(μ,0_{n×n},G), that is X has Dirac distribution in μ. Thus, \(X \sim AEC_{n,p}\!\left (\mu,0_{n \times n},g_{SMN;G}^{(0,p)}\right)\), where \(g_{SMN;G}^{(0,p)}\) is just a symbol to maintain notations. □
Proof of Corollary 4.2
By Corollary 4.1, the assertion follows from Lemma 2.2 with the specific dg \(g_{SMN;G}^{(k,p)}\). Particularly, for m∈{1,2}, \(I_{k+m}\!\left (g_{SMN;G}^{(k,p)}\right)\) is finite if and only if \(\mathbb {E}\left (V^{\frac {m}{p}}\right)\) is finite. To see this, consider
Here, we used notation \(C_{p} = \frac { p^{1\frac {1}{p}} }{ 2\Gamma \!\left (\frac {1}{p}\right) }\), two times Fubini’s theorem and changed variables \(s = \frac {r^{p}}{p} v\) with \(\frac {dr}{ds} = p^{\frac {1}{p}1} v^{\frac {1}{p}} s^{\frac {1}{p}1}\). Finally, by Lemma 2.2, the specific univariate variance component is
□
Proof
Proof of Lemma 4.2of Lemma 4.2 Let Z∼AN_{n,p}(0_{n},D) and assume Z to be independent of V. Making use of Eq. 8 and exploiting the independence of Z and V, for all \(B\in \mathfrak {B}^{n}\) and v>0,
where \(\tilde {Z} \sim N_{k,p}\). Because of \(\left (v^{\frac {1}{p}} W_{1} \sqrt {S_{1}}\right)\!\left (v^{\frac {1}{p}} W_{1} \sqrt {S_{1}}\right)^{\text {\texttt {T}}} = v^{\frac {2}{p}} D\) with \(v^{\frac {1}{p}} W_{1} \sqrt {S_{1}} \in \mathbb {R}^{n \times k}\) and \(\mathrm {rk\left (v^{\frac {1}{p}} W_{1} \sqrt {S_{1}}\right)} = k\), according to (1) with dg \(g_{PE}^{(k,p)}\), the assertion follows from
□
Before proving the general statement of Theorem 4.1, we prove the following particular one.
Lemma 5.3
Let \(X \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\). Then, X∼SMN_{k,p}(G) for the cdf G of a suitable positive random variable if and only if the function h def ined by \(h(y) = g^{(k,p)}\!\left (\sqrt [p]{y}\right)\), y∈[0,∞), is completely monotone.
Proof
Throughout this proof, let \(X \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\). If X∼SMN_{k,p}(G) for the cdf G of a suitable positive random variable, according to Corollary 4.1, \(g^{(k,p)} = g_{SMN;G}^{(k,p)}\) and
where \(C_{p} = \frac { p^{1\frac {1}{p}} }{ 2\Gamma \!\left (\frac {1}{p}\right) }\). Because of
for all \(m\in \mathbb {N}\cup \{0\}\), h is completely monotone in [0,∞). Now, let \(h = g^{(k,p)}\!\left (\sqrt [p]{\cdot }\right)\) be completely monotone on [0,∞). According to HausdorffBernsteinWidder theorem, see Widder (1946), h is representable as the LaplaceStieltjes transform of a nondecreasing function α, i.e.
and the integral converges for all 0<y<∞. Additionally, denoting,
Stieltjes integral properties yield
Thus,
Consequently, it remains to show that G defined by \(G(v) = \beta (v)  \lim \limits _{t \searrow 0}{\beta (t)}\), v>0, is the cdf of a positive random variable. Note that G is nondecreasing since α has this property. Hence,
It remains to show that \(1 = \lim \limits _{v\to \infty }{ G(v)}  \lim \limits _{t\searrow 0}{ G(t) }\). To this end, let \(\tilde {g}^{(k,p)}(z,r) = C_{p}^{k} \int \limits _{z^{1}}^{z}{ v^{\frac {k}{p}} e^{\frac {r^{p}}{p}v} \:d\beta (v) }\), 1<z<∞, denote a left and right truncated version of g^{(k,p)}. Using Fubini’s theorem, change of variables \(s=r\sqrt [p]{v}\) with \(\frac {dr}{ds}=v^{\frac {1}{p}}\) and the equality \(\omega _{k,p} \, I_{k}\!\left (g_{PE}^{(k,p)}\right) = 1\), we have
Because \(\tilde {g}^{(k,p)}(z,r)\) is a nonnegative function and g^{(k,p)}(r)=h(r^{p}), it follows \(0 \leq r^{k1} \tilde {g}^{(k,p)}(z,r) \leq r^{k1} g^{(k,p)}(r)\) for all z>1 and r>0. Furthermore, because of its structure as well as its nonnegativity, for all r>0, the function \(r^{k1} \tilde {g}^{(k,p)}(z,r)\) is monotonically increasing in variable z and converges to r^{k−1}g^{(k,p)}(r) as z→∞. Thus, the monotone convergence theorem of Beppo Levi yields the desired
Therefore, G defined by \(G(v) = \beta (v)  \lim \limits _{t \searrow 0}{\beta (t)}\), v>0, is the cdf of a positive random variable. Finally, because of
we have \(g^{(k,p)} = g_{SMN;G}^{(k,p)}\) a.e. in [0,∞) and X∼SMN_{k,p}(G). □
Before proving the general statement of Corollary 4.3, we prove the following particular one.
Corollary 5.1
Let \(X \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\) and assume that \(g^{(k,p)}\!\left (\sqrt [p]{\cdot }\right)\) is completely monotone in (0,∞)and has inverse LaplaceStieltjes transform α, \(g^{(k,p)}\!\left (\sqrt [p]{y}\right) = \int \limits _{0}^{\infty }{ e^{yt} \:d\alpha (t) }\), y>0. Then, X∼SMN_{k,p}(G) and the mixture cdf G satisf ies the representation
Moreover, the probability distribution corresponding to G is regular and has pdf f_{G} if and only if α is absolutely continuous and has pdf f_{α} where both pdfs are connected by
Proof
Proof of Corollary 5.1of Corollary 5.1 According to the second part of the proof of Lemma 5.3, on the one hand, there exists a nondecreasing function α satisfying \(g^{(k,p)}(\sqrt [p]{y}) = \int \limits _{0}^{\infty }{ e^{yt} \:d\alpha (t) }\), y>0. Since X∼SMN_{k,p}(G) for a suitable mixture cdf G, on the other hand, we have \(g^{(k,p)}(\sqrt [p]{y}) = g_{SMN;G}^{(k,p)}(\sqrt [p]{y}) = C_{p}^{k} \int \limits _{0}^{\infty }{ v^{\frac {k}{p}} e^{\frac {1}{p}yv} \:dG(v) }\), y>0. Then, changing variables \(z=\frac {1}{p}v\),
and using properties of Stieltjes integrals, it turns out that
Hence, regularity properties of probability distributions regarding to G and α are equivalent. Moreover, since f_{G} is the pdf of a positive random variable and there holds
t>0, according to the above equation involving f_{G}, it follows f_{G}(s)=0 for all s≤0 and
□
Proof
Proof of Theorem 4.1of Theorem 4.1 Let X∼SMAN_{n,p}(μ,D,G) for the cdf G of a positive random variable. Then, \(g^{(k,p)} = g_{SMN;G}^{(k,p)}\) according to Corollary 4.1 and \(g_{SMN;G}^{(k,p)}\!\left (\sqrt [p]{\cdot }\right)\) is completely monotone in [0,∞) according to Lemma 5.3. Vice versa, let X∼AEC_{n,p} (μ,D,g^{(k,p)}) with k=rk(D) and assume \(h(\cdot) = g^{(k,p)}\!\left (\sqrt [p]{\cdot }\right)\) to be completely monotone in [0,∞). Then, according to Lemma 5.3, g^{(k,p)} is the dg of a distribution from
i.e. \(\Phi _{g^{(k,p)}} = SMN_{k,p}(G)\phantom {\dot {i}\!}\) for a suitable cdf G of a positive random variable. Thus, \(X \stackrel {d}{=} \mu + W_{1}\sqrt {S_{1}} \tilde {X}\phantom {\dot {i}\!}\) where \(\tilde {X} \sim SMN_{k,p}(G)\) because of (1) and, finally, X∼SMAN_{n,p}(μ,Σ,G) because of Lemma 4.1. □
Proof
Proof of Corollary 4.3of Corollary 4.3 According to (1), for X∼AEC_{n,p} (μ,D,g^{(k,p)}) with rk(D)=k, we have
Because \(g^{(k,p)}\!\left (\sqrt [p]{\cdot }\right)\) is completely monotone in (0,∞), Corollary 5.1 yields \(\tilde {X} \sim SMN_{k,p}(G)\) as well as
where α is the inverse LaplaceStieltjes transform of \(g^{(k,p)}\!\left (\sqrt [p]{\cdot }\right)\). The relationship between the pdfs f_{G} and f_{α} follows in analogy to the second part of the proof of Corollary 5.1. □
5.4 Proofs regarding to “Scale mixed pgeneralized Gaussian processes having axisaligned fdds” section
Proof
Proof of Lemma 4.3of Lemma 4.3 Using Fubini’s theorem and changing variables \(y = v^{\frac {1}{p}} z\) with \(\frac {dy}{dz} = v^{\frac {1}{p}}\), for all \(k\in \mathbb {N}\) and r≥0, there holds
Since G is independent of k, see (9), the first factor on the right hand side of the latter equation is equal to the value of the dg \(g_{SMN;G}^{(k,p)}\) evaluated at r. Furthermore, the corresponding second factor equals 1. Thus, the assertion follows with
for all \(k\in \mathbb {N}\) and \(\left (x_{1},\ldots,x_{k}\right)^{\text {\texttt {T}}} \in \mathbb {R}^{k}\) where r=(x_{1},…,x_{k})^{T}_{p} and y=x_{k+1}. □
Proof
Proof of Theorem 4.2of Theorem 4.2 Let \(n\in \mathbb {N}\) and J={t_{1},…,t_{n}} an arbitrary subset of I having n elements. Then, \(J\in \mathcal {H}(I)\), and \(AEC_{n,p}\!\left (\mu,D,g_{SMN;G}^{(k,p)}\right)\) with μ=(m(t_{1}),…,m(t_{n}))^{T} and D= diag (S(t_{1}),…,S(t_{n})) where k=rk(D) is the fdd of the random process X corresponding to \(X_{J} = \left (X_{t_{1}},\ldots,X_{t_{n}}\right)^{\text {\texttt {T}}}\). Moreover, AN_{n,p} (0_{n},D) and \(\mathfrak {L}\left (\mu ^{(n)}+V^{\frac {1}{p}}Z_{J}\right)\) are the fdds of Z regarding to \(Z_{J} = \left (Z_{t_{1}},\ldots,Z_{t_{n}}\right)^{\text {\texttt {T}}}\) and of Y regarding to \(Y_{J} = \left (Y_{t_{1}},\ldots,Y_{t_{n}}\right)^{\text {\texttt {T}}}\), respectively. By (8) and Corollary 4.1,
for all \(n\in \mathbb {N}\) and every set \(J = \left \{t_{1},\ldots,t_{n}\right \} \in \mathcal {H}(I)\) with J=n. Thus, the random processes X and Y are equivalent meaning that they have one and the same family of fdds. □
Before we prove Theorem 4.3, we consider the following special case of it. To this end, notice that the sequence \(\left (\sigma _{g_{PE}^{(k,p)}}^{2}\right)_{k\in \mathbb {N}}\) of all univariate variance components of multivariate pgeneralized spherical Gaussian distributions equals the sequence \(\left (\sigma _{g_{SMN;G}^{(p)}}^{2}\right)_{k\in \mathbb {N}}\) with \(G=\mathbbm{1}_{(1,\infty)}\). Thus, according to the paragraph before Theorem 4.3, it is constant. Subsequently, an arbitrary element of it is denoted by \(\sigma _{g_{PE}^{(p)}}^{2}\phantom {\dot {i}\!}\) and satisfies \(\sigma _{g_{PE}^{(p)}}^{2} = \sigma _{g_{SMN;\mathbbm{1}_{(1,\infty)}}^{(p)}}^{2} = p^{\frac {2}{p}} \frac { \Gamma \!\left (\frac {3}{p}\right) }{ \Gamma \!\left (\frac {1}{p}\right) }\).
Lemma 5.4
Let Z={Z_{t}}_{t∈I}∼AGP_{p}(m,S). Then, Z is a second order random process, its expectation function is equal to m, and its covariance function \(\Gamma \colon I \times I \to \mathbb {R}\) is given by
The proof of this lemma follows immediately from Corollaries 4.1 and 4.2 and is therefore omitted, here.
Proof
Proof of Theorem 4.3of Theorem 4.3 Let Z={Z_{t}}_{t∈I}∼AGP_{p}(0_{I},S) be independent of V∼G. Then, according to Theorem 4.2, X is equivalent to the random process \(Y = \left \{ m(t) + V^{\frac {1}{p}} Z_{t} \right \}_{t \in I}\) and \(V^{\frac {1}{p}}\) and Z_{t} as well as \(V^{\frac {2}{p}}\) and Z_{s}Z_{t} are independent for all indices s,t∈I. Because of
and \(\mathbb {E}(Z_{t}) = 0\) for all t∈I according to Lemma 5.4, the value of expectation of X_{t} exists and is equal to m(t) if \(\mathbb {E}(V^{\frac {1}{p}})\) is finite. Furthermore, for all t∈I, the independence \(V^{\frac {2}{p}}\) and \(Z_{t} Z_{t} = Z_{t}^{2}\) yields
As Z is a second order random process, X is a second order random process, too, if \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is finite. In this case, for all s,t∈I, using the independence of \(V^{\frac {2}{p}}\) and Z_{s}Z_{t} as well as the covariance function of a centered pgeneralized Gaussian process Z having axisaligned fdds with scale function S from Lemma 5.4, it follows
The equation \(\mathbb {E}\left (V^{\frac {2}{p}}\right) \sigma _{g_{PE}^{(p)}}^{2} = \sigma _{g_{SMN;G}^{(p)}}^{2}\phantom {\dot {i}\!}\) yields the asserted result. □
Proof
Proof of Theorem 4.4of Theorem 4.4 Let X be strictly stationary. Then, for all t_{1}∈I and \(h \in H_{t_{1}} = \{ h\in \mathbb {R} \colon t_{1}+h \in I\}\), the distributions SMAN_{1,p}(m(t_{1}),S(t_{1}),G) of \(X_{t_{1}}\) and SMAN_{1,p}(m(t_{1}+h),S(t_{1}+h),G) of \(X_{t_{1}+h}\) are equal. If S(t_{1})=0, the distribution of \(X_{t_{1}}\) is the univariate Dirac distribution in m(t_{1}) which can be considered to be the scale mixture of the univariate kapec Gaussian distribution with k=0, location parameter m(t_{1}) and scale parameter 0. Therefore, for all \(h \in H_{t_{1}}\), \(\mathfrak {L}\left (X_{t_{1}+h}\right)\) is the univariate Dirac distribution in m(t_{1}), too, and it follows S(t_{1}+h)=0=S(t_{1}) for all \(h \in H_{t_{1}}\). Thus, S=0_{I}. Since \(\mathfrak {L}\left (X_{t_{1}+h}\right) = SMAN_{1,p}(m(t_{1}+h),0,G)\) is defined to be the Dirac distribution in m(t_{1}+h), it follows m(t_{1}+h)=m(t_{1}) for all \(h \in H_{t_{1}}\), i.e. m is constant on I. If S(t_{1})>0, according to “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section, for all t_{1}∈I and \(h \in H_{t_{1}}\), \(\mathfrak {L}\left (X_{t_{1}}\right)\) and \(\mathfrak {L}\left (X_{t_{1}+h}\right)\) have pdfs
respectively, where \(C_{p} = \frac { p^{1\frac {1}{p}} }{ 2\Gamma \!\left (\frac {1}{p}\right) }\). Because \(\mathfrak {L}\left (X_{t_{1}}\right) = \mathfrak {L}\left (X_{t_{1}+h}\right)\), we have \(f_{X_{t_{1}}} = f_{X_{t_{1}+h}}\), too. As \(f_{X_{t_{1}}}\) and \(f_{X_{t_{1}+h}}\), \(h \in H_{t_{1}}\), are symmetric with respect to the straight lines x=m(t_{1}) and x=m(t_{1}+h), respectively, being parallel to the ordinate axis, it follows m(t_{1})=m(t_{1}+h) for all t_{1}∈I and \(h \in H_{t_{1}}\). Thus, m is constant on I. Furthermore, since \(f_{X_{t_{1}}}(m(t_{1})) = \frac {C_{p}}{\sqrt {S(t_{1})}}\) and \(f_{X_{t_{1}+h}}(m(t_{1}+h)) = \frac {C_{p}}{\sqrt {S(t_{1}+h)}}\), the identity of these pdfs implies S(t_{1})=S(t_{1}+h) for all t_{1}∈I and \(h \in H_{t_{1}}\). Thus, the constancy of S on I is shown. The other direction of this proof is omitted, here. □
Proof
Proof of Theorem 4.5of Theorem 4.5 Let assume 1). According to Theorem 4.4, the constancy of m and S yields strict stationarity of X. Moreover, according to Theorem 4.3, it follows by the existence of expectation of \(V^{\frac {2}{p}}\) that X is a second order random process having expectation function m and covariance function Γ given by \(\Gamma (t,t) = \sigma _{g_{SMN;G}^{(p)}}^{2} S(t)\) for all t∈I and Γ(s,t)=0 for all s,t∈I with s≠t. Because of m(t)=μ and S(t)=δ for all t∈I, the expectation function of X is constantly equal to μ and the covariance function of X satisfies \(\Gamma (t,t) = \sigma _{g_{SMN;G}^{(p)}}^{2} \delta \) for all t∈I and Γ(s,t)=0 for all s,t∈I with s≠t. Thus, 1) implies 2). Further, every strictly stationary second order random process is weakly stationary and the covariance function Γ of X from 2) evaluated in (s,t)∈I×I is representable as a function only depending on the difference s−t since it follows from the property 3) of function K that \(\Gamma (t,t) = \sigma _{g_{SMN;G}^{(p)}}^{2} \delta = K(0) = K(tt)\) for all t∈I as well as Γ(s,t)=0=K(s−t) for all s,t∈I with s≠t. Thus, the implication from 2) to 3) is shown. Additionally, it follows from 3) that CovX_{s},X_{t}=Γ(s,t)=0 for all s,t∈I with s≠t and \(\mathbb {E}(X_{t}) = m(t) = \mu \) as well as \(Var(X_{t}) = \Gamma (t,t) = \sigma _{g_{SMN;G}^{(p)}}^{2} \delta \) for all t∈I. Hence, assuming 3), random variables X_{t}, t∈I, are uncorrelated and have constant expectation μ and variance \(\sigma _{g_{SMN;G}^{(p)}}^{2} \delta \). Thus, 4) follows from 3). Finally, let us assume 4) to hold. According to Theorem 4.3, X is a second order random process if \(\mathbb {E}\left (V^{\frac {2}{p}}\right)\) is finite. Furthermore, because of the definition of white noise as in 4), it holds \(m(t) = \mathbb {E}(X_{t}) = \mu \) as well as \(\sigma _{g_{SMN;G}^{(p)}}^{2} S(t) = Cov(X_{t},X_{t}) = Var(X_{t}) = \sigma _{g_{SMN;G}^{(p)}}^{2} \delta \) for all t∈I. Then, m and S are constantly equal to μ and δ, respectively. Thus, the implication from 4) to 1) is shown. □
Finally, the proof of Theorem 4.6 is based on Lemma 5.6. In preparation for the proof of this lemma, we establish the following special case.
Lemma 5.5
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) with D= diag (d_{1},…,d_{n}) having nonnegative diagonal elements and positive rank k. Further, let be \(b \in \mathbb {R}^{n}\) and \(\Gamma = \text { diag }\!\left (\gamma _{1},\ldots,\gamma _{n} \right) \in \mathbb {R}^{n \times n}\) such that \(\Gamma D \Gamma = \text { diag }\!\left (\gamma _{1}^{2} d_{1},\ldots,\gamma _{n}^{2} d_{n} \right)\) has positive rank k_{Γ}≥1. Then,
Proof
Assuming \(\gamma _{m_{\epsilon }} \neq 0\) for ε=1,…,l and \(\gamma _{m_{\epsilon }} = 0\) for ε=l+1,…,n where m_{1}<m_{2}<…<m_{l} and m_{l+1}<m_{l+2}<…<m_{n}, and using notations from “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section, it follows that
where
Since γ_{η}=0 for η∈{m_{l+1},…,m_{n}}, there holds
where
η=1,…,n, and
Then, K≥1 because of rk(ΓDΓ)≥1, and \(\Gamma W_{1} \sqrt {S_{1}}\) has K columns being the product of a positive constant, a constant from \(\mathbb {R}\backslash \{0\}\) and a unit vector of \(\mathbb {R}^{k}\). Particularly, all these unit vectors differ from each other and, using the notation δ_{im} of Kronecker’s Delta, we have
Hence, \(\Gamma W_{1} \sqrt {S_{1}}\) has k−K columns being 0_{n}. For \(Y = \left (Y_{1},\ldots,Y_{k}\right)^{\text {\texttt {T}}} \sim \Phi _{g^{(k,p)}}\), it follows that
where
η=1,…,n, and the vector \(\Gamma W_{1} \sqrt {S_{1}} Y\) consists of K different components of Y. Thus, for \(B\in \mathfrak {B}^{n}\), we have
where J={j∈{1,…,k}:i_{j}∈K}. Now, let
be an enumeration of the elements of J and
where
for η=1,…,n. Then, J=K and \(\Gamma W_{1} \sqrt {S_{1}} Y \stackrel {d}{=} MZ\) for \(Z \sim \Phi _{g_{(k)}^{(K,p)}}\phantom {\dot {i}\!}\). Thus, because of rk(M)=K, it follows
Note that M can be extended to \(\Gamma W_{1} \sqrt {S_{1}}\) by adding k−K zero columns. Therefore,
and K=rk(M)=rk(MM^{T})=rk(ΓDΓ). Finally, this yields
□
Using this particular result, we prove the following more general one.
Lemma 5.6
Let X∼AEC_{n,p} (μ,D,g^{(k,p)}) with D= diag (d_{1},…,d_{n}) having nonnegative diagonal elements and rank k≥0. Further, let be \(b \in \mathbb {R}^{n}\) and \(\Gamma = \text { diag }\!\left (\gamma _{1},\ldots,\gamma _{n} \right) \in \mathbb {R}^{n \times n}\). Then,
where \(\Gamma D \Gamma = \text { diag }\!\left (\gamma _{1}^{2} d_{1},\ldots,\gamma _{n}^{2} d_{n} \right)\) and k_{Γ}=rk(ΓDΓ)≥0.
Proof
Proof of Lemma 5.6of Lemma 5.6 Let k=0, that is X∼AEC_{n,p} (μ,0_{n×n},g^{(0,p)}). Then, ΓX+b follows the Dirac distribution in Γμ+b. Using the exchangeability of g^{(0,p)} and \(g_{(0)}^{(0,p)}\), we have
If D has positive rank and Γ is assumed to satisfy k_{Γ}=rk(ΓDΓ)≥1, the assertion coincides with the result of Lemma 5.5. Finally, let D have positive rank and Γ be assumed to satisfy k_{Γ}=rk(ΓDΓ)=0 and ΓDΓ=0_{n×n}, respectively. In Analogy to the proof of Lemma 5.5 and using the same notations, the set K in (11) is empty. Then, K=0, \(\Gamma W_{1} \sqrt {S_{1}}\) consists only of zero columns, and, for \(Y \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\) and every \(B\in \mathfrak {B}^{n}\), we have
Particularly, if B={0_{n}}, it follows that
Thus, \(\Gamma W_{1} \sqrt {S_{1}} Y = 0_{n}\)Pa.s., and the stochastic representation \(\Gamma X + b \stackrel {d}{=} (\Gamma \mu + b) + \Gamma W_{1} \sqrt {S_{1}} Y\) where \(Y \sim \Phi _{g^{(k,p)}}\phantom {\dot {i}\!}\) holds according to (1), yields
or, equivalently, \(\mathfrak {L}(\Gamma X + b) = AEC_{n,p}\!\left (\Gamma \mu +b,0_{n \times n},g_{(k)}^{(0,p)}\right)\). □
Proof
Proof of Theorem 4.6of Theorem 4.6 Let be \(n\in \mathbb {N}\) and J={t_{1},…,t_{n}} an arbitrary subset of I. Moreover, let Y_{t}=γ(t)X_{t}+b(t), t∈I, and Y={Y_{t}}_{t∈I}. Then, for \(Y_{J} = \left (Y_{t_{1}},\ldots,Y_{t_{n}}\right)^{\text {\texttt {T}}}\) and \(X_{J} = \left (X_{t_{1}},\ldots,X_{t_{n}}\right)^{\text {\texttt {T}}}\), we have
where b=(b(t_{1}),…,b(t_{n}))^{T} and Γ= diag (γ(t_{1}),…,γ(t_{n})). Since
where μ=(m(t_{1}),…,m(t_{n}))^{T} and D= diag (S(t_{1}),…,S(t_{n})) with k=rk(D), making use of Lemmata 4.3 and 5.5, it follows
Thus, \(AEC_{n,p}\!\left (\Gamma \mu + b, \Gamma D \Gamma, g_{SMN;G}^{(k_{\Gamma },p)} \right)\) is the fdd of Y corresponding to Y_{J}. Finally, because of Γμ+b=([γm+b](t_{1}),…,[γm+b](t_{n}))^{T} and ΓDΓ= diag ([γ^{2}S](t_{1}),…,[γ^{2}S](t_{n})), we get
□
Discussion
In the present paper, first, kapec distributions are introduced and their properties such as stochastic representations, moments, and densitylike representations are studied. Secondly, based on the Kolmogorov existence theorem, the existence of random processes having apec fdds with arbitrary location and scale functions and a consistent sequence of dgs of pgeneralized spherical distributions is shown. Particularly, a sequence of dgs of scale mixtures of multivariate pgeneralized Gaussian distributions with one and the same mixture distribution is consistent and the corresponding processes are pgeneralizations of elliptical random processes having axisaligned fdds, see Yao (1973) and Kano (1994) for the case of p=2. Thirdly, the question is answered when an ndimensional kapec distribution with dg g^{(k,p)} is representable as a scale mixture of ndimensional kapec Gaussian distribution for a suitable mixture distribution of a positive random variable. It is established that the complete monotony of the composition h of g^{(k,p)} with the pth root function is a necessary and sufficient condition for such representation and that the inverse LaplaceStieltjes transform of h is connected to the cdf of the mixture distribution. For the particular case p=2, the univariate consideration is covered by Andrews and Mallows (1974) and the multivariate one by Lange and Sinsheimer (1993) and GómezSánchezManzano et al. (2006), respectively.
Appendix 1: Further aspects of simulations
7.1 Algorithms to simulate apec distributions
The following two algorithms to simulate X∼AEC_{n,p} (μ,D,g^{(k,p)}) are based on the two stochastic representations of X, see Lemmata 2.1 and 2.4. In both cases, let \(\tilde {X} \sim \Phi _{g^{(k,p)}}\) and use notations from “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section.
Algorithm 1 1) Generation of a random vector \(U_{p}^{(k)}\) following the kdimensional pgeneralized uniform distribution on S_{k,p}:
a) Generate \(\tilde {Z} = \left (\tilde {Z}_{1},\ldots,\tilde {Z}_{k}\right)^{\text {\texttt {T}}}\) following the kdimensional pgeneralized Gaussian distribution by generating k independent and identically univariate pgeneralized Gaussian distributed random variables \(\tilde {Z}_{1},\ldots,\tilde {Z}_{k}\).
b) Compute \(R_{\tilde {Z}} = \tilde {Z}_{p}\) and \(U_{p}^{(k)} = \frac {\tilde {Z}}{R_{\tilde {Z}}}\).
2) Generate \(R_{\tilde {X}}\) having pdf \(f_{R_{\tilde {X}}}(r) = \omega _{k,p} \, r^{k1} g^{(k,p)}(r) \; \mathbbm{1}_{[0,\infty)}(r)\), \(r\in \mathbb {R}\), and being a univariate random radius variable.
3)Compute \(\tilde {X} = R_{\tilde {X}} \, U_{p}^{(k)}\) and \(X = \mu + W_{1}\sqrt {S_{1}} \tilde {X}\).
Algorithm 2 1) Generation of the random radius and angle variables according to Lemma 2.4: Generate
a) R with \(f_{R}(r) = \omega _{k,p} \, r^{k1} g^{(k,p)}(r) \, \mathbbm{1}_{[0,\infty)}(r)\), \(r\in \mathbb {R}\),
b) Ψ_{i} with \(f_{\Psi _{i}}(\psi _{i}) = \frac {\omega _{ki,p}}{\omega _{ki+1,p}} \frac { \left (\sin (\psi _{i})\right)^{ki1} }{ \left (N_{p}(\psi _{i})\right)^{ki+1}} \, \mathbbm{1}_{[0,\pi)}(\psi _{i})\), \(\psi _{i}\in \mathbb {R}\), for i=1,…,k−2,
c) Ψ_{k−1} with \(f_{\Psi _{k1}}(\psi _{k1}) = \frac {1}{\omega _{2,p}} \frac { 1 }{ \left (N_{p}(\psi _{k1})\right)^{2}} \, \mathbbm{1}_{[0,2\pi)}(\psi _{k1})\), \(\psi _{k1}\in \mathbb {R}\).
2) Compute \(\tilde {X} = SPH_{p}^{(k)}(R,\Psi _{1},\ldots,\Psi _{k1})\) and \(X = \mu + W_{1}\sqrt {S_{1}} \tilde {X}\).
For the particular case of simulating X∼SMAN_{n,p}(μ,D,G) where the mixture cdf G is explicitly known in a closed form, the following algorithm can be used. This is based on (8) and Lemma 4.1 where \(\tilde {X} \sim SMN_{k,p}(G)\) and notations from “The class of ndimensional rankkcontinuous axisaligned pgeneralized elliptically contoured distributions” section are used.
Algorithm 3 1) Generate \(\tilde {Z} = \left (\tilde {Z}_{1},\ldots,\tilde {Z}_{k}\right)^{\text {\texttt {T}}}\) following the kdimensional pgeneralized spherical Gaussian distribution by generating k independent and identically univariate pgeneralized Gaussian distributed random variables \(\tilde {Z}_{1},\ldots,\tilde {Z}_{k}\).
2) Generate independently a univariate random variable V having cdf G.
3) Compute \(\tilde {X} = V^{\frac {1}{p}} \cdot \tilde {Z}\) and \(X = \mu + W_{1}\sqrt {S_{1}} \tilde {X}\).
7.2 Simulation of pgeneralized Student as well as pgeneralized Slash processes
According to the method described in “Simulation” section, but simulating a 201dimensional apec Studentt and Slash distributed random vector with the help of an algorithms from Appendix 7.1 instead of 201 independent univariate pgeneralized Gaussian variables, we get approximates of trajectories of pgeneralized Studentt and pgeneralized Slash processes having axisaligned fdds. Particularly, approximate realizations of AStP_{p} (0_{[0,1]},1_{[0,1]},ν) as well as of ASlP_{p} (0_{[0,1]},1_{[0,1]},ν) for ν∈{1,3,10} and \(p=\frac {1}{2}\) and p=3, respectively, are visualized in Figs. 4 and 5. Note that our considerations are restricted to location function 0_{[0,1]} and scale function S=1_{[0,1]} while the effects of varying location and scale functions are already shown in Fig. 3. Furthermore, on the one hand, notice that the height of amplitudes of the realizations of AStP_{p} (0_{[0,1]},1_{[0,1]},ν) and ASlP_{p} (0_{[0,1]},1_{[0,1]},ν), respectively, increases if p>0 decreases or ν>0 increases. On the other hand, the effects that scales of axes are highly dependent on the specific realization of a trajectory of the process have to be in mind, too.
Abbreviations
 apec:

Axisaligned pgeneralized elliptically contoured
 cdf:

Cumulative distribution function
 dg:

density generator
 fdd:

Finite dimensional distribution
 fdds:

family of finite dimensional distributions
 kapec:

Rankkcontinuous axisaligned pgeneralized elliptically contoured
 pdf:

Probability density function
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Müller, K., Richter, WD. On pgeneralized elliptical random processes. J Stat Distrib App 6, 1 (2019). https://doi.org/10.1186/s4048801900906
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Keywords
 Axisaligned pgeneralized elliptically contoured distributions
 Densitylike representation
 Kolmogorov consistency conditions
 pgeneralized spherically invariant random processes
 Scale mixtures of multivariate axisaligned pgeneralized elliptically contoured Gaussian distributions
 Simulation