Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables

Abstract A measure-of-cone representation of skewed continuous ln,p-symmetric distributions, n ∈ N, p > 0, is proved following the geometric approach known for elliptically contoured distributions. On this basis, distributions of extreme values of n dependent random variables are derived if the latter follow a joint continuous ln,p-symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new parameters of multivariate tail behavior are introduced.


Introduction
Extreme value statistics are of interest not only in probability theory and mathematical statistics, but also in many elds of natural sciences and technique. While [12] present numerous applications of extreme value statistics to physics, [43], [22], and [8] deal with application to corrosion, computer science, and engineering, respectively. Furthermore, the appearance of extreme value statistics in nance, insurance and actuarial science is dealt with for instance in [11] and [32]. For an application to reliability theory, see [26]. General introductions into and surveys over the theory and practice of extreme value distributions are presented, amongst others, in [9], [20], [13], [29], [30], [31], and [14].
Distributions of extreme value statistics of independent and identically distributed random variables (rvs) are already determined in [15]. The case of correlated rvs with a joint normal distribution is dealt with, e.g., in [17], [28] and [19]. The probability density functions (pdfs) of the maximum statistic and of linear combinations of order statistics of arbitrary absolutely continuous dependent rvs, with an emphasis on elliptically contoured sample distributions, are considered in [3] and [2]. These considerations are followed up in [18] and some paper referred to there, and further developed by representing the results with the help of skewed distributions. Earlier results dealing with this relationship can be found in [21] where the two-dimensional Gaussian case is considered. A geometric approach to bivariate and multivariate skewed elliptically contoured distributions is presented in [16] and [42], respectively, where the measure-of-cone representation of these distributions is worked out.
In [27], some steps of the development of the theory of ln,p-symmetric distributions and their applications are reviewed. An emphasis of this overview is on geometric measure representations and on a methodological study of their applications to the derivation of exact statistical distributions if samples follow a joint continuous ln,p-symmetric distribution. Such exact results extend those valid for multivariate spherically symmetric sample distributions. Notice that results on exact extreme value distributions holding if the sample distribution is an arbitrary element of the larger class of elliptically contoured distributions may be again further extended assuming a p-generalized elliptically contoured sample distribution. The latter distributions appeared already in Section 3.5 in [4] and were studied from the point of view of star-shaped distributions in [39] and from that of convex and radially concave contoured distributions in [40]. All these distribution classes provide more exibility in modeling data. For more details, we refer to Section 2.
Skewed ln,p-symmetric distributions are introduced in [4] and applied to the maximum distribution of continuous l ,p -symmetrically distributed random vectors in [6]. From a certain point of view, the aim of the present paper is to extend this result to the nite number n of dependent rvs following a joint continuous ln,psymmetric distribution. To this end, a geometric approach to skewed ln,p-symmetric distributions, following and generalizing main ideas of the measure-of-cone representations from [42] in the present situation, will be developed and, afterwards, used for the derivation of extreme value distributions. From another point of view, our present results extend, although in a slightly di erent form, those derived in [27] for three or four dependent random variables to the case of an arbitrary nite number of dependent rvs following a joint continuous ln,p-symmetric distribution. As an additional result of the present paper it becomes obvious that the explicit representations of distributions of extreme value statistics derived in the two earlier papers of the authors may be considered as representations of skewed distributions being alternatives to the known ones.
In order to represent maximum distributions in terms of skewed distributions, there are two basic approaches. On the one hand, the authors of [6] present a full-length proof of transforming the result on the maximum pdf from [26] directly into the language of skewed distributions. To this end, they start from the Laplace and Gaussian cases, respectively, and extend the results, passing the two-dimensional p-power exponential case, stepwise in quick succession to the l ,p -symmetric case with an arbitrary density generator (dg). On the other hand, the authors of [16] and [42] deduce a certain measure-of-cone representation of skewed elliptically contoured distributions.
If one compares the numerous applications of geometric measure representation done so far in the literature, one may distinguish between the direct and more advanced applications. To roughly de ne these terms, direct applications deal with immediate calculations of the so called intersection-proportion function of a particular random event under consideration, and more advanced applications deal with types of intersection-proportion functions being typical for whole classes of random events. While the small sample studies in the earlier papers of the authors belong to the direct type of applications of the geometric measure representation, the present paper deals with a more advanced one in Section 5.2.
The rest of the present paper is organized as follows. In Section 2, general information about the considered class of ln,p-symmetric distributions and the corresponding class of skewed distributions are given. Based on the ln,p-symmetric model assumption, in Section 3, the cumulative distribution function (cdf) and the pdf of extreme value statistics for a nite number of dependent rvs are considered. The density of the maximum is graphically illustrated for several choices of dgs of the l ,p -symmetric sample distributions and for certain values of the shape/ tail parameter p > . Studying the asymptotic behavior of the pdf of the maximum statistic of the components of an n-dimensional p-generalized Gaussian distributed random vector is another aim of Section 3. In Section 4.1, the gures of the maximum pdf of three rvs following a joint l ,p -symmetric Pearson Type VII or Kotz type distribution are discussed w.r.t. the heaviness of their tails. The tail index and two new parameters describing the tail behavior of some ln,p-symmetric distributions, and the centers of l , -symmetric Kotz type distributions are considered in Sections 4.2 and 4.3, respectively. In Section 5, rst, a geometric measure-of-cone representation of skewed ln,p-symmetric distributions is introduced leading afterwards to an advanced geometric method of proof. Second, based upon this, the results of Section 3 are proved and the advanced geometric method of proof is concisely compared to the direct one from [27]. In Section 6, some conclusions are drawn from the results of the present paper. Appendix A provides density generators of and some more basic facts on subclasses of ln,p-symmetric distributions. Appendix B deals with the in uence which parameters of density generators have onto the heaviness or lightness of multivariate distribution tails.

Preliminaries
In this section, the model class of ln,p-symmetric distributions and the class of skewed ln,p-symmetric distributions are introduced and some of their basic properties are recalled. Throughout this paper, let p > be arbitrary but xed and let |x|p = n k= |x k | p p , x = (x , . . . , xn) T ∈ R n , denote the p-functional which is a norm if p ≥ and, according to [25], an antinorm if p ∈ ( , ). A function g : ( , ∞) → ( , ∞) satisfying the assumption < I(g) < ∞ is called a density generating function (dgf) of an n-variate distribution where An random vector X : Ω → R n de ned on a probability space (Ω, A, P) and having the pdf is said to follow the continuous ln,p-symmetric distribution with dgf g where ωn,p = Γ p n p n− Γ n p denotes the ln,p-generalized surface content of the ln,p-unit sphere Sn,p = {x ∈ R n : |x|p = }, see [37]. Further, a dgf g of a continuous ln,p-symmetric distribution meeting the condition I(g) = ωn,p is called dg of this distribution, and denoted by g (n) . We denote the cdf and the pdf of the corresponding distribution by Φ g (n) ,p and φ g (n) ,p , respectively, where For examples of dgs, for references to the literature on ln,p-symmetric distributions, as well as for a discussion of a further aspect of notation, we refer to [27] and the Appendix A, respectively.
Remark 1. According to [37], an ln,p-symmetrically distributed random vector X with dg g (n) satis es the stochastic representation where U (n) p is n-dimensional p-generalized uniformly distributed on the ln,p-unit sphere Sn,p, R and U (n) p are stochastically independent and R is a nonnegative random variable with pdf f R (r) = ωn,p r n− g (n) (r), r > . (4) Here and in what follows X d = Z and X ∼ Ψ means that the random vectors X and Z follow the same distribution law and that the random vector X follows the distribution law Ψ, respectively. Moreover, let In be the n × n unit matrix and n the zero vector in R n .
Remark 2. The density of the nonnegative random variable R p is a g (n) , p -generalization of the χ -density, Having in mind the slight change of notation from [39] and [27], respectively, this distribution law was de ned for arbitrary dgf g in [37] having pdf and was considered earlier in [35,36] for the particular cases p = and g (n) = g (n) PE , respectively.
Remark 3. Let X ∼ Φ g (n) ,p . According to [4], Consequently, the components of X are uncorrelated. The components of X are only independent if g (n) = g (n) PE , see Remark 5 and Appendix A.
As announced in Section 1, now, we further discuss the connection between ln,p-symmetric and standardized multivariate elliptically contoured (or spherical) distributions. As it is well known according to [7], the Euclidean stochastic representation of an n-dimensional with parameter vector µ ∈ R n and nonnegative de nite matrix Σ ∈ R n×n having rank k elliptically contoured random vector Z is given by where U (k) is uniformly distributed on the unit sphere in R k ,R ≥ is independent of U (k) , Σ = AA T is a rank factorization of Σ, and the cdf F of R is connected to the characteristic generator ϕ of Z by where Ω k denoted the characteristic function of U (k) . In contrast to (5), a non-Euclidean stochastic representation of Z is given for regular Σ by where R and U are independent, R follows a g-generalized χ -distribution which is de ned in [35], and U follows the ellipsoidal or |·| a -generalized uniform probability distribution introduced in De nition 3.2 in [38]. A generalization of (6) is given in Section 4.7 in [39] where p-generalized elliptically contoured distributions are considered. De nition 8 in Section 4.4 of the same paper deals with a stochastic representation of even more general star-shaped distributions.
It is well known that some elliptical distributions can be obtained as normal variance mixtures. For the general case, see, e.g., [24], and for the particular case of the multivariate skew-t distribution, see [10]. In [5], a scale mixture expression of exponential power distributions is given for a generalized-t distribution de ned in [23]. A general scale mixture of p-generalized normal distributions is dealt with in Section 3.3 in [4] including the special case of p-generalized Student-t distribution.
For getting a rst impression of the heaviness of the multivariate tails of ln,p-symmetric distributions one may study typical values of the quantile function Q g (n) ,p : [ , ] → R w.r.t. the domain Bn,p = x ∈ R n : |x| p ≤ de ned by According to (3) and Remark 2, where F R has the density (4). In Appendix B, the quantiles Q g (n) ,p (q) are computed for several values of n, di erent dgs and q ∈ { . , . , . , . , .

}.
According to the work of [21], [18], [6], and other authors, distributions of extreme value statistics are intrinsically connected with certain skewed versions derived from the considered sample distributions. Skewed versions of ln,p-symmetric distributions are studied in [4]. To follow these authors, let X = X ( ) T , X ( ) T T be a random vector having a continuous l k+m,p -symmetric distribution with dg g (k+m) where X ( ) : Ω → R k and X ( ) : Ω → R m . We recall that, di ering from (2), the density of X was represented in [4] as g (k+m) (|x| p p ). Taking, here and later, this minor change of notation into account, the dg g (k) (k+m) of the marginal distribution of X ( ) in R k satis es Furthermore, for Λ ∈ R m×k , Γ = (Λ, −Im), and Σ = ΓΓ T = Im + ΛΛ T , the cdf of ΓX will be denoted by F ( ) m,p x; Σ, g (m) (k+m) , x ∈ R m . Moreover, for every x ( ) ∈ R k , the conditional density of X ( ) given X ( ) = x ( ) is ,p . Let Y be a random vector following this distribution, ,p , then its cdf is A k-dimensional random vector Z having a pdf of the form is said to follow the skewed l k,p -symmetric distribution SS k,m,p Λ, g (k+m) with dimensionality parameter m, dg g (k+m) and skewness/ shape matrix-parameter Λ. Further, the parameter k is called the co-dimensionality parameter and the cdf of Z is denoted by F k,m,p ·; Λ, g (k+m) .
The following remark deals with the e ects of interchanging columns or rows in the matrix-parameter Λ and is proven in Section 5.1. i.e. SS k,m,p Λ, g (k+m) = SS k,m,p M Λ, g (k+m) where M Λ arises from Λ by interchanging rows.
According to [4], skewed l k,p -symmetric distributions are constructed via selection mechanisms from ln,psymmetric distributions. Particularly, if X ( ) : Ω → R k and X ( ) : Ω → R m are again two subvectors of a random vector X, X ∼ Φ g (k+m) ,p , then L X ( ) X ( ) < ΛX ( ) = SS k,m,p Λ, g (k+m) (9) where L(Y) denotes the distribution law of the random vector Y. Therefore, part b) of Remark 4 re ects the exchangeability of the components of an ln,p-symmetrically distributed random vector within the skewed l k,p -symmetrical distributions.

Extreme value distributions for arbitrary nite sample sizes
We recall that speci c results for exact distributions of order statistics of up to three and extreme value statistics up to four dependent rvs following a joint continuous ln,p-symmetric distribution, n ∈ { , , }, are proved in earlier papers of the authors by directly applying the geometric measure representation of ln,psymmetric distributions. In Section 3.1, exact extreme value distributions are derived if an arbitrary nite number of dependent rvs follows a continuous ln,p-symmetric distribution. To this end, an advanced geometric method of proof will be developed in Section 5 following main ideas for geometrically representing skewed elliptically contoured distributions in [42]. In Section 3.2, our results are graphically visualized. On the one hand, gures of densities are drawn for the special case of jointly trivariate -generalized Gaussian distributed rvs and, on the other hand, for the case of three dependent rvs following a joint l ,p -symmetric Kotz type and a joint l ,p -symmetric Pearson Type VII distribution, respectively. Another aim of this section is to provide an idea of the asymptotic behavior of the maximum pdf for n-variate p-generalized Gaussian sample distribution as n tends to in nity.

. Dimension and co-dimension representations
The results of this section re ect strong connections between skewed distributions and distributions of extremes. Such type of connection can already be seen in [21], [18], and in papers of several other authors. The present results are derived based upon the geometric measure representation in [37]. This representation applies directly if only small numbers of dependent rvs are considered. In [6], the particular result on the maximum pdf of two dependent rvs being jointly l ,p -symmetrically distributed is transformed directly into the typical representation of skewed distributions. Here, we use the geometric representation of ln,p-symmetric measures in a more advanced way. To be more concrete, we derive from it a measure-of-cone representation of skewed l k,p -symmetric distributions in Corollary 2, Section 5. Let E (ν) denote the ν × (n − ν) matrix whose st column is ν = ( , . . . , ) T ∈ R ν and whose remaining n − ν − columns are ν-dimensional zero vectors.
,p , for every ν ∈ { , . . . , n − }, the cdf Fn:n of the maximum statistic of the components of X satis es the representation For arbitrary n ∈ N and p > , Theorem 1 provides numerous representations of the cdf of the maximum statistic from ln,p-symmetrically distributed populations in terms of skewed distributions. In particular, these are alternatives to that given in Theorem 1 in [26] for n = and Theorems 1 and 3 in [27] for n = and n = , respectively. In the case of n = , the equivalence of these two alternative representations is shown in [6] by direct integral transformation. Furthermore, in the speci c case p = and ν = n − , the result of Theorem 1 is covered by Theorem 7 in [18]. Now, we brie y discuss the impact of the parameter ν ∈ { , . . . , n − } in Theorem 1. Recalling that Fn−ν,ν,p t n−ν; E (ν) , g (n) is the cdf of SSn−ν,ν,p E (ν) , g (n) and considering the construction of skewed ln−ν,psymmetric distributions via select mechanisms again, X ( ) and X ( ) in equation (9) are real-valued (n − ν)and ν-dimensional subvectors of an ln,p-symmetrically with dg g (n) distributed random vector X, respectively. Hence, the parameter ν de nes the dimensionality of the conditioning subvector in equation (9)  ; I + T , g ( ) = = ν+ and equation (10) reads as Fn:n(t) = F n− , ,p t n− ; e (n− ) T , g (n) , t ∈ R. Therefore, maximum distributions for continuous ln,p-symmetric vectors may particularly be represented as skewed l n− ,p -symmetric distributions. For all the other parameters ν ∈ { , . . . , n − }, the matrix Iν + ν T ν is not diagonal. Due to this, the normalizing con- (n) is not as easy to handle with as in the case ν = and, in general, its exact value is unknown. Nevertheless, all representations of the maximum cdf given in (10) are well treatable since the corresponding maximum pdfs, see Corollary 1, do not depend on the mentioned normalizing constant.
Vice versa, it may be sometimes of interest to read equation (10) in a reverse order, meaning that for every ν ∈ { , . . . , n − }, the cdf of Z, Z ∼ SSn−ν,ν,p(E (ν) , g (n) ), at the particular argument t n−ν, t ∈ R, satis es where Σ = Iν + ν T ν and Fn:n denotes the cdf of the maximum statistic of the components of X, X ∼ Φ g (n) ,p . Using the direct application of the geometric measure representation shortly discussed in Section 1, the cdf Fn:n is already determined for n ∈ { , , } in earlier papers of the authors. Thus, by substituting Fn:n on the right hand side of equation (11) for n ∈ { , , } by these previous representations, one gets alternative representations of the cdf Fn−ν,ν,p t n−ν; E (ν) i , g (n) , t ∈ R, to that following from Section 2 as the componentwise de ned integral of the pdf of SSn−ν,ν,p(E (ν) i , g (n) ) over the region z ∈ R n−ν : z < t n−ν . Summarizing this section up to here, there are n − equivalent possibilities of representing the maximum cdf Fn:n using skewed distributions. This e ect also occurs in the representation of the maximum pdf fn:n as it can be seen in the following corollary being proved in Section 5.2.
Using the general relation between maximum and minimum statistics if the sample distribution is symmetric and continuous, and the functions provided by Theorem 1 and Corollary 1, the minimum cdf and pdf of the components of X, X ∼ Φ g (n) ,p , are given by F :n (t) = − Fn:n(−t) and f :n (t) = fn:n(−t), respectively. In particular, for ν = n − , this yields , t ∈ R.
In due consideration of the mentioned slight variation of notation for ln,p-symmetric densities, formula (12) equals for n = the result in [6]. It is worthwhile to note that the structure of our results on extremes of several dependent variables in (12) and (13) is similar to that of the corresponding results of the pdf of the maximum and the minimum statistic, respectively, of n independent and identically distributed rvs, see [9].
φp(s) ds, t ∈ R, denote the pdf and the cdf of the onedimensional marginal distribution, respectively. The pdfs (12) and (13) of the extreme value statistics of the components of X simplify to fn: respectively. Note that, in these speci c cases, our representations (12) and (13) also follow from [9] since the components of p-generalized Gaussian distributed random vectors are independent.

. Visualization of the maximum density
In the present section, the pdf of the maximum statistic of dependent, jointly ln,p-symmetrically distributed rvs is illustrated for some choices of the dg g (n) and the shape/ tail parameter p > . Figures 1-3  Kt;M,β,γ and g (n) PT ;M,ν of these subclasses of the ln,p-symmetric distributions are given in the Appendix A. Note that the present choice of the shape/ tail parameter p and of the parameters appearing in the de nitions of the dgs coincides with that in Figures 2 and 3 in [27]. Thus, one can compare the graphs of the pdf of the maximum statistics presented here with that of the median statistic drawn there.
Another aim of this section is to give a visual impression of the asymptotic behavior of the pdf of the maximum statistic for increasing sample sizes. This will be done in n-dimensional p-generalized normally distributed populations, i.e. in the case of independent components. Using the dg g (n) PE and the four choices of the parameter p as in Figures 4 and 5, the impact of an increasing sample size onto the shape of the maximum pdf is re ected in Figure 6. Furthermore, in this gure, one can perceive the impact of parameter p which is, on the one hand, a shape parameter and, on the other hand, a tail parameter since the shape of the multivariate density level sets depends on it and the tail pdf of the underlying random vector becomes lighter if p increases, respectively.
Note that the axes in Figures 4, 5 and 6 are scaled di erently, and that both the left and the right hand sides of Figures 4(a), 5(a), 5(b) and 6(a) show a black graph as a respective benchmark. Moreover, note that illustrations of the minimum pdf can be received if the graphs of the corresponding maximum pdfs are mirrored at the ordinate axis.

Tails and centers of maximum distributions
In this section, we review some of the gures of Section 3.2 in detail. Moreover, we give additional information on the tail index and on heaviness and on lightness of tails of ln,p-symmetric distributions.

. Light, heavy and extremely far tails
The in uence that the parameter ν > of an l , -symmetric Pearson Type VII distribution with parameter M = has onto the heaviness of the tails of the median distribution of three dependent rvs is discussed in Section 4 of [27]. In the present section, rst, an analog study for the case of the maximum distribution in such populations is done. Second, we examine the heaviness of the tails of the maximum distribution from other l ,p -symmetric sample distributions with respect to their parameters.
Note that the graphs of the right hand side of Figure 5(a) convey the impression that the visualized densities build a monotonically decreasing sequence of functions. The more detailed views in Figure 7, however,          (d) Figure 7: Some detailed views on the graphs of the right hand side of Figure 5(a).
three dependent rvs following a joint l , -symmetric distribution with dg g ( ) PT ; ,ν if ν increases. The same tendency can be seen in the case of l , -symmetric Pearson Type VII sample distribution with constant parameter M = and increasing parameter ν. In Table 1, the integral A p M,ν (z , z ) = For the sake of comparison of heaviness of the tails of the median and the maximum distribution, our present consideration is for the class of l , -symmetric Pearson Type VII distributions with the same di erent values of ν > and M = as in Section 4 in [27]. Equally, interpreting the values in Table 1, one can   Table 1, all the three cases in Figure 5(a) cover only a small part of the entire probability mass. Therefore, the behavior of the graphs outside the considered interval [− . ; . ] is not clearly predictable. However, for the case of ν = , Figure 8 shows the pdf of the maximum statistic of rvs following a joint l , -symmetric distribution with dg g ( ) PT ; , over the interval [− ; ], suggesting a monotonically increasing behavior over the negative real line and a monotonically decreasing one over the positive real line. Note that only an extremely small proportion of probability mass generates a peak of the density function close right to the zero point and that the overwhelming part of probability mass is seemingly uniformly distributed on an extremely long interval. Additionally, in Table 2, the integral A p M, (z , z ) is numerically evaluated for p = and M ∈ { , , } and p = and M ∈ , , , respectively and several asymmetric real intervals [z , z ] where "≈ " denotes the case that the value of A p M, (z , z ) rounded to the sixth decimal place equals . These values emphasize a decreasing heaviness of the tails of the maximum distribution in jointly l ,p -symmetrically Pearson Type VII distributed samples if the parameter M > p increases and the parameter ν > is constant.
Finally, Table 3, shows di erent heaviness of the tails of the distribution of the maximum statistic of three jointly l , -symmetric Kotz type distributed rvs for the choices of parameters from Figure 4

. Tail indices
While the tail of the distribution of the univariate maximum statistic in jointly l ,p -symmetrically Pearson Type VII or Kotz type distributed samples is explored in Section 4.

. . Heavy Tails
Let us call a random vector regularly varying with tail index α > w.r.t. the p-functional |·| p , p > , if there exist a positive constant α and a probability law S on the Borel-σ-eld B n ∩ Sn,p of subsets of the |·| p -unit sphere such that for every x > x α µz ·; |·| p ⇒ S as z → ∞ where the symbol ⇒ means weak convergence, and Furthermore, S is called the spectral measure w.r.t. |·| p . Note that this complies with the common notion of a regularly varying distribution w.r.t. the p-norm if p ≥ . Now, let X ∼ Φ g (n) ,p with dg g (n) . Thus, because of the stochastic representation (3), i.e., according to the notion in [33], if the survival function of the univariate random variable R is regularly varying at ∞ with index −α, then the n-dimensional p-generalized uniform distribution on Sn,p is the spectral measure of Φ g (n) ,p w.r.t. |·| p , Example 1. In the particular case g (n) = g (n) PT ;M,ν with M > n p and ν > , the application of L'Hôpital's rule yields Consequently, the ln,p-symmetric Pearson Type VII distribution with parameters M > n p and ν > has tail index Mp − n. In the case n = and p ≥ , this result is already covered by Example 6 in [41].

. . Light Tails
Adopting de Haan's notion of Γ-variation, see [33] and original references cited therein, let us call a random vector Γ-varying w.r.t. the p-functional |·| p , p > , if there exist a positive function f and a probability law S on B n ∩ Sn,p such that for every x ∈ R Here, f is called an auxiliary function. If X ∼ Φ g (n) ,p then and hence S is the p-generalized uniform distribution on the Borel-σ-eld over Sn,p if i.e. the survival function of the univariate p-radius variable R of X is Γ-varying with auxiliary function f .
Example 2. We consider the special case of an ln,p-symmetric Kotz type distribution with parameters M > − n p , β > and γ > , i.e. g (n) = g (n) Kt;M,β,γ , and denote the pdf and the cdf of the Radius variable R by f Kt and F Kt , respectively. For f (z) = βγp z pγ− , z > , and for every x ∈ R, from of the asymptotic equivalence relations Thus, a random vector following an ln,p-symmetric Kotz type distribution with parameters M > − n p , β > and γ > is Γ-varying w.r.t. the p-functional and with auxiliary function f (z) = z −pγ βγp , z > . Remark 7. For a random vector being ln,p-symmetrically Pearson Type II distributed with parameter ν > one can verify neither the property of regular variation nor that of Γ-variation w.r.t. the p-functional, p > , since the support of pdf of the corresponding radius variable is bounded.

. . Bounded supports
Let X be an ln,p-symmetrically contoured random vector such that the distribution of |X| p has a bounded support, and let us denote the right endpoint of if by x E . If there exist a positive constant α and a probability law S on B n ∩ Sn,p such that for every x > then we call the random vector X bounded regularly varying with tail index α > w.r.t. |·| p , p > . In this case, and S is the n-dimensional p-generalized uniform measure on Sn,p if for every x > i.e. if the survival function of the univariate random variable x E −R is regularly varying at ∞ with index −α.
Example 3. In the special case g (n) = g (n) PT ;ν with ν > , on the one hand, x E = and, on the other hand, for all x > . Therefore, the ln,p-symmetric Pearson Type II distribution with parameter ν > or a random vector following that distribution is bounded regularly varying with tail index ν + w.r.t. |·| p .

. Light and heavy distribution centers
While extremely long concentration intervals and extremely far tails of probability distributions were studied in Sections 4.1 and 4.2, here the focus is on the centers of l , -symmetric Kotz type distributions for certain choices of parameters. Nevertheless, it is worthwhile to mention that ln,p-symmetric Kotz type distributions have relatively light tails caused by the exponential part of their dgs. The monomial part of the dg ensures that the heaviness of the distribution center can be modeled with the help of the parameter M. It can be seen from Figure 9(a), that the choice of M = is the decisive factor to have a heavy distribution center. Standard examples of this distributional type are power exponential and, particularly, Gaussian and Laplace distributions, and their ln,p-generalizations. Additionally, in the case M = , Figure 9(a) shows that the parameter β > controls mainly the height and the parameter γ > mainly the decay behavior of the dg. In the case M = , the parameter β as well as γ regulate the height of the dg. Furthermore, they induce a shift of the probability mass. The e ect of these choices of parameters on the shape of the pdf of the maximum distribution of three dependent rvs following a joint l , -symmetric Kotz type distribution can be seen in Figure 9(b). Finally, one can compare Figure 9(b) with Figure 4 to get an impression of the impact of a further increase of the parameter p.

Proofs . Measure-of-cone representations of skewed l k,p -symmetric distributions
An initial step of the proof of Theorem 1 deals with deriving a representation of the cdfs of skewed l k,psymmetric distributions with dimensionality parameter m in terms of speci c ln,p-symmetric measures of 0    (b) The pdf of the maximum statistic for the same parameters as in Figure 9(a). cones by analogy to what was done in [42] for skewed elliptically contoured distributions with dimensionality parameter m = . Afterwards, this measure-of-cone representation is used to prove Remark 4. Let a random vector Z follow the skewed l k,p -symmetric distribution with dimensionality parameter m, dg g (k+m) , and matrix-parameter Λ ∈ R m×k , and let e (n) j , j = , . . . , n, still denote the jth standard unit vector of R n . For arbitrary z ∈ R k , we consider the cone Cm (a , , . . . , a ,m , a , . . . , a k where the quantities a ,l = −Γ T e (m) l , l = , . . . , m, a i = e (k+m) i , i = , . . . , k, and Γ = (Λ, −Im) are as in Section 2. This cone generalizes that in [42] where the case m = is dealt with. Note that A (z) has its vertex at z T , (Λz) T T . Furthermore, A (z) is the intersection of k + m half spaces, m of whom containing the origin in its boundary.
If Z ∼ SS k,m,p Λ, g (k+m) , then the cdf of Z allows the representation where F ( ) m,p ·; Im + ΛΛ T , g (m) (k+m) is the normalizing constant from (8).
. Then where the integral z −∞ h(ζ ) dζ , z ∈ R k , is de ned as a k-fold one. Further, using the notation X = X ( ) , X ( ) T where the vectors X ( ) and X ( ) take their values in R k and R m , respectively, Here, φ g (k+m) ,p is the pdf of X, and Thus, Note that Theorem 1 in [42] follows from Lemma 1 for m = and p = .
Let Op denote the ln,p-generalized surface content on B n ∩ Sn,p and Fp : ( , ∞) → [ , ∞) the ln,p-sphere intersection-proportion function (ipf) de ned by According to [37], but with suitably adapted notations as in [39,40] and [27], for arbitrary p > and n ∈ N, the continuous ln,p-symmetric distribution Φ g (n) ,p with dg g (n) satis es the geometric measure representation Note that this formula was rst proved for p = and g (n) = g (n) PE in [34], and for p = and arbitrary dgf g in [35].
Using formula (15) and the symmetry of both Op and the ln,p-unit sphere Sn,p, the following lemma provides some invariance properties of the ln,p-symmetric probability measure.
Let D be an n × n sign matrix, i.e. D = diag{d , . . . , dn} with d i ∈ { , − }, i = , . . . , n. D is an orthogonal matrix and, by analogous considerations as before, Consequently, Φ g (n) ,p is a member of the class SI of sign invariant distributions, considered in [1]. Note that one can prove Lemma 2 alternatively without using formula (15), starting from Φ g (n) ,p (A) = A φ g (n) ,p (x) dx = A g (n) (|x|p) dx and using the invariance properties of the p-functional | · |p.
Next, we are going to generalize the speci c measure-of-cone representation formula for the cdf of a skewed l k,p -symmetrically distributed random vector given in Lemma 1. Doing this, rst, a class of cones is introduced and, then, the permutation and sign invariance of the measure Φ g (k+m) ,p is basically utilized. We recall that the invariance of Φ g (n) , with respect to all orthogonal transformations was used to construct general geometric measure representations for n = in [16] and for arbitrary n in [42].
Let Proof of Remark 4. Initializing the proof of part a), let M be a k×k permutation matrix and Z ∼ SS k,m,p Λ, g (k+m) .
Since |M ξ |p = |ξ |p, ξ ∈ R k , and | det(M )| = , To prove part b), let M be a permutation matrix in R m and Z ∼ SS k,m,p M Λ, g (k+m) . As in the proof of Note that = F k,m,p z; Λ, g (k+m) , z ∈ R k .

. Applying the advanced geometric method to extremes
We recall that advanced applications of the geometric measure representation (15) make use of types of intersection-percentage functions (14) being valid for whole classes of random events. The classes of events considered here are cones generated by intersecting half spaces. In this section, the measure-of-cone representations from Corollary 2 are mainly used to prove Theorem 1 which, in turn, is basic to establish Corollary 1 and Remark 5. Moreover, the direct and the advanced geometric methods considered in this paper are brie y compared. Let A n n (t) = (x , . . . , xn) T ∈ R n : x < t, . . . , xn < t , t ∈ R, be a sublevel set generated by the maximum statistic of an n-dimensional random vector X = (X , . . . , Xn) T . Then, Moreover, illustrations of the set A n n (t) may be found in the two earlier papers of the authors for n ∈ { , }.
Proof of Theorem 1. Let ν ∈ { , . . . , n − } be xed. Further, letB ( ) is a disjoint decomposition of the cone A (t) with vertex at (t, t) T , see Figure 10. Similarly, on the one hand, one has that A (t) = B ( ) (t) ∪B ( ) , (t) where On the other hand, using this and the decomposition of A (t) again, into ν + cones so that each of them, which is an intersection of n half spaces from R n , contains the origin in the boundary of ν of its n intersecting half spaces. In the spherical case p = , this idea, that at least one hyperplane contains the origin, arises in [16] for n = , and in [42] for an arbitrary n. Indicating the topological interior of the set A ⊆ R n by int(A), i.e. one can transform each of the conesB (ν) i,j (t), j = , . . . , ν, by permutation into the cone B (ν) i (t), where the matrix M i,j ∈ R n×n de nes the transposition σ i,j with σ i,j (i) = j, σ i,j (j) = i and σ i,j (l) = l for all l ∈ [ , n]\{i, j}. By Lemma 2, for every i ∈ { , . . . , n − ν}, the maximum cdf Fn:n of the components of a continuous ln,psymmetrically distributed random vector X with dg g (n) satis es Fn:n(t) = P(X ∈ A n n (t)) =Φ g (n) ,p (A n n (t)) Further, is a matrix whose ith column is ν and all the others are ν-dimensional zero vectors. Then, Corollary 2 implies for every ν ∈ { , . . . , n − } and i ∈ { , . . . , n − ν} that Furthermore, on the one hand, On the other hand, as the above cdf Fn−ν,ν,p( · ; E (ν) i , g (n) ) is evaluated at a point whose components are all equal to each other, Remark 4 a) yields for every i , i ∈ { , . . . , n − ν}, where Z ∼ SSn−ν,ν,p(E (ν) i , g (n) ) and the matrix M i ,i ∈ R (n−ν)×(n−ν) is the transposition matrix as before. Hence, without any loss of generality, the parameter i in (16) can be chosen as i = .
By evaluating the cdf of the distribution law SSn−ν,ν,p E (ν) i , g (n) at a point whose components equal each other and using the exchangeability of the components of X ( ) , the alternative proof of choosing i = without loss of generality is nished.
Summarizing the methods used in the present paper and that used in [27], there were two ways of applying the geometric measure representation (15) to get exact distribution formulae: the direct and the advanced one. The latter opens the possibility to turn over to considering conditional distributions and their densities.
The case ν = n − can be dealt with in an analogous way.

Discussion
In the present paper, we study exact distributions of order statistics under nonstandard model assumptions.
The dependence of the variables considered here is caused in the interplay of the dg and the shape/ tail parameter of the multivariate sample distribution. As in the most known spherical case p = , the uncorrelatedness of the considered rvs leads to their independence if and only if the dg of the sample vector distribution is that of the p-generalized multivariate Gaussian distribution. Because of the possible arbitrary choice of p, p > , our considerations are not restricted to sample vector densities being convex contoured as in [41] but include cases where density level sets are radially concave w.r.t. the standard fan in R n , see [39,40]. We have established an advanced geometric method using measure-of-cone representations for deriving extreme value distributions of jointly continuous ln,p-symmetrically distributed dependent rvs.
In particular, both the rst and the second order moments of this distribution do not exist.
A. The l n,p -symmetric Pearson Type II distribution Let   Q g (n) ,p q = . q = . q = . q = . q = .

B Tables for domain quantiles of l n,p -symmetric distributions w.r.t. the l n,p -unit ball
Here, we compute values of the quantile function Q g (n) ,p at the points q ∈ { . , . , . , . , .
g (n) = g (n) ν ∈ , , , and g (n) PT ;ν with ν ∈ , , . Remember that the ln,p-symmetric power exponential distribution with parameter γ = is the n-variate p-generalized Gaussian distribution and that the ln,p-symmetric Student-t distribution with parameter ν = is the ln,p-symmetric Cauchy distribution.