Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results

Abstract

In this paper, the representability of the generalized Student’s distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws.

1. Introduction

1.1. History of the Problem and Motivation

The attention to the Student, Burr and Snedecor–Fisher distributions is motivated by the fact that that they have power-type decreasing tails and possess very simply representable densities and/or distribution functions. These circumstances make these models very convenient for the statistical analysis of the regularities observed in data with non-normal leptokurtic and heavy-tailed empirical distributions. The reasons for the analytical and asymptotical study of the Student distribution have already been discussed in [1]. The Burr distribution (more exactly, Burr XII distribution) was first introduced analytically in [2] and then together with its special cases such as Lomax distribution, Fisk distribution, etc. (see [3] or [4], Sect. 7.3 and 6.4.1) proved to be a successful model for the size distribution of incomes and other econometric and actuarial problems (see, e.g., [5,6]). The analytic properties of the Burr distribution, as well as its relation with some other distributions, are discussed in [7]. At the same time, the Snedecor–Fisher distribution is widely applied in statistics as the auxiliary distribution in testing hypotheses, e.g., concerning the scale parameters of normally distributed data. In descriptive statistics this distribution is used considerably more rarely. One of the very few examples of the applications of this distribution is wireless communication systems. In particular, in recent papers [8,9], this distribution is used to model effects related to the turbulence-induced fading in free space optical communication systems.

This paper can be regarded as a continuation and extension of [1], where, among other results, normal scale mixture representations were obtained for the generalized Student distribution. In the present paper, the representability of the generalized Student distribution as uniform and normal-scale mixtures is considered. We introduce the generalized Burr distribution as the distribution of the exponentiated random variable with the generalized Lomax distribution (for the definition of the latter, see [1]). It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws.

These results can provide theoretical grounds for the use and utility of the asymmetric generalized Student distribution, generalized Burr distribution and its symmetric and asymmetric two-sided versions, as well as the Snedecor–Fisher distribution together with its two-sided versions. Briefly, these grounds are that these distributions can be limiting in rather general and simple limit theorems for random sums or extreme order statistics in samples with random sizes and, therefore, can serve as asymptotic approximations for statistical regularities inherent in stopped-random-walk-type or lifetime-type inhomogeneous data.

The paper is organized as follows. In Section 1.2, auxiliary definitions and results are collected. Here, we also introduce the generalized Burr distribution. In Section 2, mixture properties of the generalized Burr and Snedecor–Fisher distributions are discussed. In Section 3, the representability of the generalized Student and generalized Burr distributions in the form of uniform scale mixtures is discussed and some related results are presented. In Section 4, we introduce the two-sided generalized Burr distribution and discuss the asymmetric generalizations of the generalized Student and generalized Burr distributions. Section 4.1 deals with ‘randomized’ asymmetrization, whereas in Section 4.2, we discuss the normal variance–mean mixture approach to the construction of asymmetric generalizations. In Section 5, we demonstrate that the generalized Burr, asymmetric two-sided generalized Burr, and asymmetric generalized Student distributions can be limit laws in limit theorems for statistics constructed from samples with random sizes. In Section 5.1, limit theorems for random sums are proven in which the generalized Burr distribution is a limit law. Section 5.2 contains limit theorems for extreme order statistics in samples with random sizes in which the generalized Burr distribution is a limit law. Finally, in Section 5.1 we prove two versions of the central limit theorem for random sums in which the limit laws are the asymmetric two-sided generalized Burr and asymmetric generalized Student distributions.

1.2. Auxiliary Definitions and Notation

We will assume that the random variables under consideration are defined on one and the same probability space $(\mathsf{\Omega },\mathcal{A},\mathsf{P})$.

The symbol ∘ will denote the product of independent random elements. By ⟹ and $\stackrel{d}{=}$, we will denote the convergence in distribution and coincidence of distributions, respectively. The end of the proof will be marked by the symbol □. We will use the notation ${\mathbb{I}}_A\left(z\right)$ for the indicator function of a set A: ${\mathbb{I}}_A\left(z\right)=1$ if $z\in A$; otherwise, ${\mathbb{I}}_A\left(z\right)=0$.

A random variable with the standard exponential distribution will be denoted as $W_1$:

$$\mathsf{P}(W_1<x)=\left[1-e^{-x}\right]{\mathbb{I}}_{[0,\infty )}\left(x\right).$$

For $r>0$, Euler’s gamma function will be denoted as $\mathsf{\Gamma }\left(r\right)$:

$$\mathsf{\Gamma }\left(r\right)={\int }_0^{\infty }{z}^{r-1}{e}^{-z}dz.$$

A gamma-distributed random variable with shape parameter $r>0$ and scale parameter $\lambda >0$ will be denoted as $G_{r,\lambda }$,

$$\mathsf{P}(G_{r,\lambda }<x)={\int }_0^{x}g(z;r,\lambda )dz,\mathrm{with}g(x;r,\lambda )=\frac{{\lambda }^r}{\mathsf{\Gamma }\left(r\right)}{x}^{r-1}{e}^{-\lambda x}{\mathbb{I}}_{[0,\infty )}\left(x\right),$$

Obviously, a random variable with the standard exponential distribution can be denoted in two ways: $G_{1,1}=W_1$.

The absolutely continuous distribution defined by the density

$$g{g}_{r,\alpha ,\mu }\left(x\right)=\frac{\left|\alpha \right|{\mu }^r}{\mathsf{\Gamma }\left(r\right)}{x}^{\alpha r-1}{e}^{-\mu x^{\alpha }}{\mathbb{I}}_{[0,\infty )}\left(x\right)$$

(1)

where $\alpha \in \mathbb{R}$, $\mu >0$, $r>0$ is called the generalized gamma distribution is. A random variable with the density $g{g}_{r,\alpha ,\mu }\left(x\right)$ will be denoted as ${\overline{G}}_{r,\alpha ,\mu }$. It can be easily seen that

$${\overline{G}}_{r,\alpha ,\mu }\stackrel{d}{=}{G}_{r,\mu }^{1/\alpha }\stackrel{d}{=}{\mu }^{-1/\alpha }{G}_{r,1}^{1/\alpha }\stackrel{d}{=}{\mu }^{-1/\alpha }{\overline{G}}_{r,\alpha ,1}.$$

(2)

Let $\gamma >0$. The distribution of the random variable $W_{\gamma }$:

$$\mathsf{P}\left(W_{\gamma }<x\right)=\left[1-e^{-x^{\gamma }}\right]{\mathbb{I}}_{[0,\infty )}\left(x\right),$$

is called the Weibull distribution with shape parameter $\gamma$. It is easy to see that

$$W_{\gamma }\stackrel{d}{=}{W}_1^{1/\gamma }\stackrel{d}{=}{\overline{G}}_{1,\gamma ,1}.$$

(3)

The random variable $W_{\alpha }^{-1}$ is said to have the inverse Weibull or Fréchet distribution:

$$\mathsf{P}(W_{\alpha }^{-1}<x)=\mathsf{P}(W_{\alpha }\ge \frac{1}{x})=exp\left\{x^{-\alpha }\right\},x\ge 0.$$

By X, we denote a random variable with the standard normal distribution:

$$\mathsf{P}(X<x)\stackrel{\mathrm{def}}{=}\mathsf{\Phi }\left(x\right)={\int }_{-\infty }^x\phi \left(z\right)dz,\phi \left(x\right)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2},x\in \mathbb{R}.$$

The probability density of the strictly stable law with characteristic exponent $\alpha$ and parameter $\theta$ will be denoted by $s_{\alpha ,\theta }\left(x\right)$. This distribution corresponds to the characteristic function

$${\mathfrak{g}}_{\alpha ,\theta }\left(t\right)=exp\left\{-{\left|t\right|}^{\alpha }exp\left\{-\frac{i\pi \theta \alpha }{2}\mathrm{sign}t\right\}\right\},t\in \mathbb{R},$$

(4)

with $0<\alpha \le 2$, $\left|\theta \right|\le {\theta }_{\alpha }=\min \{1,\frac{2}{\alpha }-1\}$ (see, e.g., [10]). A random variable with a characteristic function (4) will be denoted as $S_{\alpha ,\theta }$. It is easily seen that $S_{2,0}\stackrel{d}{=}\sqrt{2}X$.

The values of the parameters $\theta =1$ and $0<\alpha \le 1$ correspond to strictly stable distributions concentrated on the non-negative half line. The values $\alpha =1$ and $\theta =\pm 1$ correspond to the stable distributions degenerate in $\pm 1$, respectively. All other strictly stable distributions are absolutely continuous. In general, the densities $s_{\alpha ,\theta }\left(x\right)$ cannot be explicitly represented in terms of elementary functions. There are only four exceptions: the normal distribution ($\alpha =2$, $\theta =0$), the Cauchy distribution ($\alpha =1$, $\theta =0$), the Lévy distribution ($\alpha =\frac{1}{2}$, $\theta =1$) and the distribution symmetric to the Lévy law ($\alpha =\frac{1}{2}$, $\theta =-1$). In [11,12], one can find expressions for stable densities in terms of generalized Meijer G functions (Fox functions).

Lemma 1

([13]). If $\frac{1}{2}\le \delta <\infty$ and $0<\alpha \le 1$, then

$$\mathsf{E}{S}_{\alpha ,1}^{-\delta }=\frac{\mathsf{\Gamma }\left(\frac{\delta }{\alpha }\right)}{\alpha \mathsf{\Gamma }\left(\delta \right)}.$$

(5)

Let $a,b\in \mathbb{R}$, $a<b$. A random variable with the uniform distribution on $[a,b]$ will be denoted by $R_{[a,b]}$.

Let $\alpha >0$. By the symmetric exponential power distribution, we will mean the absolutely continuous distribution defined by its probability density

$$p_{\alpha }\left(x\right)=\frac{\alpha }{2\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)}·e^{-{\left|x\right|}^{\alpha }},-\infty <x<\infty .$$

(6)

Hereinafter, for the sake of simplicity, a single parameter $\alpha$ in representation (6) will be used since this parameter is characteristic and determines the shape of distribution (6). A random variable with probability density $p_{\alpha }\left(x\right)$ will be denoted as $Q_{\alpha }$, and the $\alpha =1$ relation (6) defines the Laplace distribution with zero mean and variance 2:

$$p_1\left(x\right)=\frac{1}{2}{e}^{-\left|x\right|},x\in \mathbb{R}.$$

If $\alpha =2$, then relation (6) defines the normal (Gaussian) distribution with zero mean and variance $\frac{1}{2}$:

$$\sqrt{2}{Q}_2\stackrel{d}{=}X.$$

(7)

Exponential power distributions (6) were introduced by M. T. Subbotin in [14]. For more details concerning the properties of exponential power distributions, see [13,15] and the references therein.

It is easy to verify that

$$|Q_{\alpha }{|}^{\alpha }\stackrel{d}{=}{G}_{1/\alpha ,1}.$$

(8)

Since some properties of exponential power distributions will be used in our further reasoning, we formulate them as the following lemmas.

Lemma 2

([13]). Let $\alpha \in (0,2]$, ${\alpha }^{\prime }\in (0,1]$. Then

$$Q_{\alpha {\alpha }^{\prime }}\stackrel{d}{=}{Q}_{\alpha }\circ U_{\alpha ,{\alpha }^{\prime }}^{-1/\alpha },$$

(9)

where $U_{\alpha ,{\alpha }^{\prime }}$ is a r.v. such that: if ${\alpha }^{\prime }=1$, then $U_{\alpha ,{\alpha }^{\prime }}=1$ for any $\alpha \in (0,2]$ and if $0<{\alpha }^{\prime }<1$, then $U_{\alpha ,{\alpha }^{\prime }}$ is absolutely continuous with probability density

$$u_{\alpha ,{\alpha }^{\prime }}\left(x\right)=\frac{{\alpha }^{\prime }\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)}{\mathsf{\Gamma }\left(\frac{1}{\alpha {\alpha }^{\prime }}\right)}·\frac{s_{{\alpha }^{\prime },1}\left(x\right)}{x^{1/\alpha }}·{\mathbb{I}}_{(0,\infty )}\left(x\right).$$

Corollary 1

([16]). Any symmetric exponential power distribution with $\alpha \in (0,2]$ is a scale mixture of normal laws:

$$Q_{\alpha }\stackrel{d}{=}\sqrt{\frac{1}{2}{U}_{2,\alpha /2}^{-1}}\circ X.$$

Corollary 2

(e.g., see [13]). Any symmetric exponential power distribution with $\alpha \in (0,1]$ is a scale mixture of Laplace laws:

$$Q_{\alpha }\stackrel{d}{=}{U}_{1,\alpha }^{-1}\circ Q_1.$$

Lemma 3

([13]). For any $\alpha \in (0,1]$, the distribution of the random variable $U_{2,\alpha /2}^{-1}$ is mixed exponential:

$$U_{2,\alpha /2}^{-1}\stackrel{d}{=}4U_{1,\alpha }^{-2}\circ W_1.$$

Corollary 3

([13]). For any $\alpha \in (0,1]$, the distribution of the random variable $U_{2,\alpha /2}^{-1}$ is infinitely divisible.

Lemma 4

([17,18]). For any $\alpha \in (0,2]$

$$Q_{\alpha }\stackrel{d}{=}{R}_{[-1,1]}\circ G_{1+1/\alpha ,1}^{1/\alpha }.$$

Lemma 4, in combination with (7), implies the following uniform scale mixture representation for the normal distribution.

Corollary 4

([17,18]). The normal distribution is a scale mixture of uniform distributions:

$$X\stackrel{d}{=}{R}_{[-1,1]}\circ G_{3/2,2}^{1/2}.$$

Actually, Corollaries 1, 2 and Lemma 4 obviously imply the following general statements.

Corollary 5.

I. 

Any scale mixture of the exponential power distributions is a uniform scale mixture. In particular, any normal scale mixture is a uniform scale mixture. Any mixed exponential distribution is a scale mixture of the uniform distribution on $[0,1]$.

II. 

If the distribution of a random variable ξ is symmetric so that the distribution of $\left|\xi \right|$ is mixed exponential, then the distribution of ξ is a normal scale mixture

In [19], it was shown that any gamma distribution with a shape parameter no greater than one is a mixed exponential. Namely, the following statement holds.

Lemma 5

([19]). Let $r\in (0,1]$. Then, the density $g(x;r,\mu )$ of a gamma distribution can be represented as

$$g(x;r,\mu )={\int }_0^{\infty }z{e}^{-zx}p(z;r,\mu )dz,$$

where

$$p(z;r,\mu )=\frac{{\mu }^r}{\mathsf{\Gamma }(1-r)\mathsf{\Gamma }\left(r\right)}·\frac{{\mathbb{I}}_{[\mu ,\infty )}\left(z\right)}{{(z-\mu )}^{r}z}.$$

(10)

Furthermore, if $r>1$, then the gamma distribution cannot be represented as a mixed exponential distribution.

Lemma 6

([20]). Let $r\in (0,1)$, $\mu >0$ and $G_{r,1}$ and $G_{1-r,1}$ be independent gamma-distributed random variables. Then, the density $p(z;r,\mu )$ defined by (10) corresponds to the random variable

$$Z_{r,\mu }=\frac{\mu (G_{r,1}+G_{1-r,1})}{G_{r,1}}\stackrel{d}{=}\mu Z_{r,1}\stackrel{d}{=}\mu \left(1+\frac{1-r}{r}{R}_{1-r,r}\right),$$

(11)

where $R_{1-r,r}$ is the r.v. with the Snedecor–Fisher distribution defined by the probability density

$$f(x;1-r,r)=\frac{{(1-r)}^{1-r}{r}^r}{\mathsf{\Gamma }(1-r)\mathsf{\Gamma }\left(r\right)}·\frac{{\mathbb{I}}_{(0,\infty )}\left(x\right)}{x^r[r+(1-r)x]}.$$

(12)

In other words, if $r\in (0,1)$, then

$$G_{r,\mu }\stackrel{d}{=}{W}_1\circ Z_{r,\mu }^{-1}.$$

(13)

Lemma 7

([13]). Let $\alpha \ge 1$. Then, the one-sided exponential power distribution is a scale mixture of the Weibull distributions:

$$|Q_{\alpha }|\stackrel{d}{=}{Z}_{1/\alpha ,1}^{-1/\alpha }\circ W_{\alpha }.$$

(14)

Lemma 8

([21]). If $\gamma \in (0,1)$, then the Weibull distribution with parameter γ is mixed exponential:

$$W_{\gamma }=W_1\circ S_{\gamma ,1}^{-1}.$$

(15)

The commonly used Student distribution was introduced in [22] as the distribution of the random variable

$$T_r\stackrel{d}{=}X\circ G_{r,r}^{-1/2},$$

where $r>0$ is the shape parameter usually called ‘the number of degrees of freedom’. The probability density of the Student distribution, up to scale and location transformation, has the form

$$f_r\left(x\right)=\frac{\mathsf{\Gamma }(r+\frac{1}{2})}{\sqrt{\pi r}\mathsf{\Gamma }\left(r\right)}{\left(1+\frac{x^2}{r}\right)}^{-(r+1/2)},x\in \mathbb{R}.$$

The term Lomax distribution, also called the Pareto Type II distribution (see [3]) corresponds to the probability distribution, up to scale and location transformation, defined by the density of the form

$$f_r^{*}\left(x\right)=\frac{r}{{(1+x)}^{r+1}},x\ge 0,$$

where $r>0$ is the shape parameter.

Let $r_1>0$, $r_2>0$. The distribution of the random variable

$$P_{r_1,r_2}=G_{r_1,r_1}\circ G_{r_2,r_2}^{-1}.$$

is called the Snedecor–Fisher distribution with parameters $(r_1,r_2)$ (or Fisher f-distribution). The probability density $p_{r_1,r_2}\left(x\right)$ of the Snedecor–Fisher distribution has the form

$$p_{r_1,r_2}\left(x\right)=\frac{r_1^{r_1}{r}_2^{r_2}}{\mathrm{B}(r_1,r_2)}·\frac{x^{r_1-1}}{{(r_{1}x+r_2)}^{r_1+r_2}}.$$

Here and in what follows, $\mathrm{B}(a,b)$ is the beta function:

$$\mathrm{B}(a,b)=\frac{\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(b\right)}{\mathsf{\Gamma }(a+b)},a>0,b>0.$$

It is easy to see that

$$p_{1,r}\left(x\right)=\frac{r^r}{\mathrm{B}(1,r)}·\frac{1}{{(x+r)}^{r+1}}=\frac{r^r\mathsf{\Gamma }(r+1)}{\mathsf{\Gamma }\left(r\right)}·\frac{1}{{(x+r)}^{r+1}}=\frac{r^{r+1}}{{(x+r)}^{r+1}}=\frac{1}{{(\frac{x}{r}+1)}^{r+1}},$$

that is, the Snedecor–Fisher distribution with parameters $1,r$ coincides with the Lomax distribution up to the change in the scale parameter $x\mapsto \frac{x}{r}$.

Let $\alpha \in (0,2]$, $r\in \mathbb{R}$ and $\alpha r\ge 1$. Let the random variables $Q_{\alpha }$ and $G_{r,r}$ be independent. The distribution of the random variable

$$T_{r,\alpha }\stackrel{\mathrm{def}}{=}{Q}_{\alpha }\circ G_{r,r}^{-1/\alpha }.$$

(16)

is called the generalized Student with parameters $\alpha$ and r. The probability density function $f_{r,\alpha }\left(x\right)$ of $T_{r,\alpha }$ has the form

$$f_{r,\alpha }\left(x\right)=\frac{\alpha }{2r^{1/\alpha }\mathrm{B}(r,\frac{1}{\alpha })}{\left(1+\frac{{\left|x\right|}^{\alpha }}{r}\right)}^{-(r+1/\alpha )},x\in \mathbb{R}.$$

(17)

If $\alpha =2$, then the generalized Student distribution turns into the classical Student distribution up to the re-parametrization. And if, in addition, $r=1$, then the generalized Student distribution is the Cauchy distribution.

If $\alpha =1$, the generalized Student distribution turns into the two-sided Lomax distribution.

The generalized Student distribution is representable as a normal scale mixture, and not only in the obvious case $\alpha =2$, as is stated in the following lemma implied by Corollary 1.

Lemma 9

([1]). Let $\alpha \in (0,2]$. Then

$$T_{r,\alpha }\stackrel{d}{=}\sqrt{D_{r,\alpha }}\circ X,$$

(18)

where

$$D_{r,\alpha }\stackrel{d}{=}\frac{1}{2}{U}_{2,\alpha /2}^{-1}\circ G_{r,r}^{-2/\alpha }.$$

(19)

For the properties of the random variable $D_{r,\alpha }$, see [1].

The distribution of the random variable

$$|T_{r,\alpha }|\stackrel{d}{=}\left|Q_{\alpha }\right|\circ G_{r,r}^{-1/\alpha }$$

can be called generalized Lomax. In general, with arbitrary $\alpha \in (0,2]$, the distribution of $|T_{r,\alpha }|$ can be called folded generalized Student or one-sided generalized Student distribution. However, the term generalized Lomax distribution is conventional. It can be easily seen that the distribution of the random variable $|T_{r,1}|$ corresponds to the Lomax distribution.

From (17), it is easy to see that the probability density $f_{r,\alpha }^{*}\left(x\right)$ of the generalized Lomax distribution has the form

$$f_{r,\alpha }^{*}\left(x\right)=\frac{\alpha }{r^{1/\alpha }\mathrm{B}(r,\frac{1}{\alpha })}{\left(1+\frac{x^{\alpha }}{r}\right)}^{-(r+1/\alpha )},x\ge 0.$$

(20)

Recall that here, $\alpha \in (0,2]$ and $r>0$ so that $\alpha r>1$.

Lemma 10

([1]). For $\alpha \in (0,1]$ and $r>\frac{1}{\alpha }$, the generalized Lomax distribution is the mixed exponential:

$$|T_{r,\alpha }|\stackrel{d}{=}{\left(U_{1,\alpha }\circ G_{r,r}^{1/\alpha }\right)}^{-1}\circ W_1.$$

(21)

Let $r>0$, $\gamma >0$. The Burr type XII distribution (or simply the Burr distribution) is defined by its density

$$b_{r,\gamma }\left(x\right)=r\gamma \frac{x^{\gamma -1}{\mathbb{I}}_{[0,\infty )}\left(x\right)}{{(1+x^{\gamma })}^{r+1}},x\in \mathbb{R}.$$

Let $\gamma >0$. Consider the distribution of the random variable $|T_{r,1}{|}^{1/\gamma }$. For $x\ge 0$, we have $\mathsf{P}\left(\right|T_{r,1}{|}^{1/\gamma }<x)=\mathsf{P}(|T_{r,1}|<x^{\gamma })$, so that

$$\frac{d}{dx}\mathsf{P}\left(\right|T_{r,1}{|}^{1/\gamma }<x)=\frac{r\gamma x^{\gamma -1}}{\gamma {(1+x^{\gamma })}^{r+1}}\equiv b_{r,\gamma }\left(x\right).$$

In other words, the Burr distribution corresponds to the random variable $|T_{r,1}{|}^{1/\gamma }$, that is, to the exponentiated Lomax-distributed random variable.

By analogy, for $\alpha \in (0,2]$, $r>0$, $\gamma >0$, the distribution of the exponentiated random variable with the generalized Lomax distribution, $|T_{r,\alpha }{|}^{1/\gamma }$, will be called the generalized Burr distribution. With the account of (20), it can be easily seen that the density $b_{r,\alpha ,\gamma }^{*}\left(x\right)$ of the generalized Burr distribution has the form

$$b_{r,\alpha ,\gamma }^{*}\left(x\right)=\frac{\alpha \gamma x^{\gamma -1}}{r^{1/\alpha }\mathrm{B}(r,\frac{1}{\alpha })}{\left(1+\frac{x^{\alpha \gamma }}{r}\right)}^{-(r+1/\alpha )},x\ge 0.$$

(22)

Recall that here, $\alpha \in (0,2]$ and $r>0$ so that $\alpha r>1$.

2. Mixture Properties of the Generalized Burr and Snedecor–Fisher Distributions

The generalized Burr distribution inherits some favourable analytic properties of the generalized Lomax distribution described, say, in [1].

Proposition 1.

For $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma \in (0,1]$, the generalized Burr distribution is a mixed exponential:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}{Y}_{r,\alpha ,\gamma }\circ W_1,$$

where

$$Y_{r,\alpha ,\gamma }\stackrel{d}{=}{\left(U_{1,\alpha }\circ G_{r,r}^{1/\alpha }\right)}^{-1/\gamma }\circ S_{\gamma ,1}^{-1}.$$

Proof. 

The proof consists of the sequential use of Lemmas 10 and 8 with the account of (3). □

Corollary 6.

For $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma \in (0,1]$, the generalized Burr distribution is infinitely divisible.

Proof. 

This statement follows from Proposition 1 and the result of [23] stating that the product of two independent non-negative random variables is infinitely divisible, if one of the two is exponentially distributed. □

Corollary 7.

For $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma \in (0,1]$, the generalized Burr distribution is a scale mixture of the folded normal distribution:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}\sqrt{Y_{r,\alpha ,\gamma }^{*}}\circ \left|X\right|,$$

where

$$Y_{r,\alpha ,\gamma }^{*}=2Y_{r,\alpha ,\gamma }^2\circ W_1,$$

and the random variable $Y_{r,\alpha ,\gamma }$ is defined in Proposition 1.

Proof. 

The statement follows from Proposition 1 with the account of the easily verified relation $W_1\stackrel{d}{=}\sqrt{2W_1}\circ \left|X\right|$. □

Now, relax the condition $\gamma \in (0,1]$ used in the statements above.

Recall that the random variable $W_{\alpha }^{-1}$ is said to have the inverse Weibull or Fréchet distribution:

$$\mathsf{P}(W_{\alpha }^{-1}<x)=\mathsf{P}(W_{\alpha }\ge \frac{1}{x})=exp\left\{x^{-\alpha }\right\},x\ge 0.$$

Proposition 2.

I. 

Let $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma >0$. Then, the generalized Burr distribution is a scale mixture of the Weibull distribution with parameter γ:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}{\left(U_{1,\alpha }\circ G_{r,r}^{1/\alpha }\right)}^{-1/\gamma }\circ W_{\gamma }.$$

(23)

II. 

Let $1<\alpha \le 2$, $r>\frac{1}{\alpha }$ and $\gamma >0$. Then, the generalized Burr distribution is a scale mixture of the Weibull distribution with parameter $\alpha \gamma$:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}{\left(Z_{1/\alpha ,1}\circ G_{r,r}\right)}^{-1/\left(\alpha \gamma \right)}\circ W_{\alpha \gamma }.$$

(24)

III. 

Let $1<\alpha \le 2$, $\frac{1}{\alpha }<r<1$ and $\gamma >0$. Then, the generalized Burr distribution is a scale mixture of Fréchet distribution with parameter $\alpha \gamma$:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}{\left(Z_{r,1}\circ G_{1/\alpha ,r}\right)}^{1/\left(\alpha \gamma \right)}\circ W_{\alpha \gamma }^{-1}.$$

(25)

Proof. 

Part I trivially follows from Lemma 10. Part II follows from the definition of the generalized Burr distribution, that of the generalized Student distribution and Lemma 7 used in the numerator of the fraction defining the generalized Student distribution. Part III follows from the definition of the generalized Burr distribution, that of the generalized Student distribution and Lemma 7 used in the denominator of the fraction defining the generalized Student distribution. □

It should be noted that the representation of the usual Burr distribution as a scale mixture of the Weibull distribution with the generalized gamma mixing distribution of the scaling factor was obtained in [24].

From relation (8), it follows that

$$|T_{r,\alpha }|\stackrel{d}{=}\left|Q_{\alpha }\right|\circ G_{r,r}^{-1/\alpha }\stackrel{d}{=}{\left(G_{1/\alpha ,1}\circ G_{r,r}^{-1}\right)}^{1/\alpha }\stackrel{d}{=}{\left(\frac{1}{\alpha }{G}_{1/\alpha ,1/\alpha }\circ G_{r,r}^{-1}\right)}^{1/\alpha }\stackrel{d}{=}{\alpha }^{-1/\alpha }{P}_{1/\alpha ,r}^{1/\alpha }.$$

(26)

Hence, we obtain one more representation for the generalized Burr distribution, this time via the Snedecor–Fisher distribution:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}{\alpha }^{-1/\alpha \gamma }{P}_{1/\alpha ,r}^{1/\alpha \gamma }.$$

(27)

At the same time, from (27) with $\alpha \stackrel{\mathrm{def}}{=}1/r_1$ and $r\stackrel{\mathrm{def}}{=}{r}_2$ we obtain the representation of the Snedecor–Fisher distribution via the generalized Burr distribution: if $r_1\ge 1$ and $r_2>r_1$, then

$$P_{r_1,r_2}\stackrel{d}{=}{r}_1^{-1}{\left|T_{r_2,1/r_1}\right|}^{1/r_1},$$

(28)

that is, up to the scaling parameter, the Snedecor–Fisher distribution with parameters $r_1\ge 1$ and $r_2>r_1$ coincides with the generalized Burr distribution with parameters $r=r_2$, $\alpha =1/r_1$ and $\gamma =r_1$. Hence, Propositions 1, 2 and Corollaries 5 and 6 can be re-written in terms of the Snedecor–Fisher distribution. The corresponding statements will be formulated as corollaries to the results mentioned above.

Proposition 1 together with Lemma 6 imply the following result.

Corollary 8.

I. 

If $r_1\in (0,1]$, then the Snedecor–Fisher distribution is mixed exponential for any $r_2>0$:

$$P_{r_1,r_2}\stackrel{d}{=}{(r_1{Z}_{r_1,1}\circ G_{r_2,r_2})}^{-1}\circ W_1,$$

where the random variable $Z_{r_1,1}$ was defined in Lemma 6.

II. 

If $r_1>1$, then the Snedecor–Fisher distribution is mixed exponential for $r_2>r_1$:

$$Y_{r_2,1/r_1,r_1}\circ W_1,$$

where the random variable $Y_{r_2,1/r_1,r_1}$ was defined in Corollary 1.

Corollary 9.

The Snedecor–Fisher distribution is infinitely divisible if $r_1\in (0,1]$ or if $r_1>1$ and $r_2>r_1$.

Corollary 10.

For $r_1\ge 1$ and $r_2>r_1$, the Snedecor–Fisher distribution is a scale mixture of the folded normal distribution:

$$P_{r_1,r_2}\stackrel{d}{=}{r}_1^{-1}\sqrt{Y_{r_2,1/r_1,r_1}^{*}}\circ \left|X\right|,$$

where

$$Y_{r_2,1/r_1,r_1}^{*}=2Y_{r_2,1/{\mathbb{R}}_1,r_1}^2\circ W_1,$$

and the random variable $Y_{r_2,1/r_1,r_1}$ was defined in Proposition 1.

Corollary 11.

I. 

Let $r_1\ge 1$ and $r_2>r_1$. Then, the Snedecor–Fisher distribution is a scale mixture of the Weibull distribution with parameter $r_1$:

$$P_{r_1,r_2}\stackrel{d}{=}{\left(r_1{G}_{r_2,r_2}\right)}^{-1}\circ U_{1,1/r_1}^{-1/r_1}\circ W_{r_1}.$$

II. 

Let $\frac{1}{2}\le r_1<1$ and $r_1<r_2<1$. Then, the Snedecor–Fisher distribution is a scale mixture of the standard Fréchet distribution:

$$P_{r_1,r_2}\stackrel{d}{=}\left(Z_{r_2,1}\circ G_{r_1,r_2}\right)\circ W_1^{-1}.$$

(29)

3. Uniform Scale Mixture Representations for the Generalized Student and Generalized Burr Distributions and Related Results

First, consider the possibility of representation of the generalized Student distribution as a scale mixture of uniform distributions. For more details concerning the properties of uniform scale mixtures and their statistical applications, see [25,26,27].

Proposition 3.

Let $\alpha \in (0,2]$, $r>0$. Then

$$T_{r,\alpha }\stackrel{d}{=}{\left(\frac{1+\alpha }{\alpha }\right)}^{1/\alpha }{R}_{[-1,1]}\circ P_{1+1/\alpha ,r}^{1/\alpha }.$$

Proof. 

The desired statement follows from Lemma 4 and the definition of the Snedecor–Fisher distribution:

$$T_{r,\alpha }\stackrel{\mathrm{def}}{=}{Q}_{\alpha }\circ G_{r,r}^{-1/\alpha }\stackrel{d}{=}{R}_{[-1,1]}\circ G_{1+1/\alpha ,1}^{1/\alpha }\circ G_{r,r}^{-1/\alpha }\stackrel{d}{=}$$

$$\stackrel{d}{=}{\left(\frac{1+\alpha }{\alpha }\right)}^{1/\alpha }{R}_{[-1,1]}\circ {\left[G_{1+1/\alpha ,1+1/\alpha }\circ G_{r,r}^{-1}\right]}^{1/\alpha }\stackrel{d}{=}{\left(\frac{1+\alpha }{\alpha }\right)}^{1/\alpha }{R}_{[-1,1]}\circ P_{1+1/\alpha ,r}^{1/\alpha }.$$

□

From Lemmas 2 and 4, we can obtain the following result which can be regarded as a kind of ‘multiplication theorem’ for gamma distributions.

Let $\alpha \in (0,2]$, ${\alpha }^{\prime }\in (0,1]$. From Lemma 4, we have

$$Q_{\alpha {\alpha }^{\prime }}\stackrel{d}{=}{R}_{[-1,1]}\circ G_{1+1/\left(\alpha {\alpha }^{\prime }\right),1}^{1/\left(\alpha {\alpha }^{\prime }\right)}.$$

(30)

At the same time, Lemma 4 predated by Lemma 2 yields

$$Q_{\alpha {\alpha }^{\prime }}\stackrel{d}{=}{U}_{\alpha ,{\alpha }^{\prime }}^{-1/\alpha }\circ Q_{\alpha }\stackrel{d}{=}{U}_{\alpha ,{\alpha }^{\prime }}^{-1/\alpha }\circ R_{[-1,1]}\circ G_{1+1/\alpha ,1}^{1/\alpha }.$$

(31)

In [28], it was demonstrated that scale mixtures of symmetric uniform distributions are identifiable; that is, if

$$R_{[-1,1]}\circ V_1\stackrel{d}{=}{R}_{[-1,1]}\circ V_2$$

for some non-negative random variables $V_1$ and $V_2$, then $V_1\stackrel{d}{=}{V}_2$ (see example II in [28]). This means that relations (30) and (31) imply the following statement.

Proposition 4.

For any $\alpha \in (0,2]$ and ${\alpha }^{\prime }\in (0,1]$

$$G_{1+1/\alpha {\alpha }^{\prime },1}\stackrel{d}{=}{\left[U_{\alpha ,{\alpha }^{\prime }}^{-1}\circ G_{1+1/\alpha ,1}\right]}^{{\alpha }^{\prime }}.$$

Corollary 12.

For any $\gamma \ge \frac{3}{2}$, we have

$$G_{\gamma ,1}\stackrel{d}{=}{\left[U_{2,1/(2\gamma -2)}^{-1}\circ G_{3/2,1}\right]}^{1/(2\gamma -2)}.$$

Roughly speaking, Corollary 9 states that any gamma distribution with a shape parameter no less than 3/2 can be represented as a tempered scale mixture of the gamma distribution with shape parameter 3/2; the tempering is meant as the power transform of the argument of the distribution function.

As an example of the application of Proposition 4, consider the following statement presenting a kind of ‘multiplication theorem’ for the Snedecor–Fisher distributions.

For $\alpha \in (0,2]$, introduce the function $k\left(\alpha \right)=1+1/\alpha$.

Corollary 13.

Let $\alpha \in (0,2]$, $\beta \in (0,2]$, ${\alpha }^{\prime }\in (0,1]$. Then

$$P_{k\left(\alpha {\alpha }^{\prime }\right),k\left(\beta {\alpha }^{\prime }\right)}\stackrel{d}{=}{\left(U_{\beta ,{\alpha }^{\prime }}\circ U_{\alpha ,{\alpha }^{\prime }}^{-1}\circ P_{k\left(\alpha \right),k\left(\beta \right)}\right)}^{{\alpha }^{\prime }}.$$

As regards the generalized Burr distribution, Lemma 4 and Proposition 1 imply the following representation.

Proposition 5.

Let $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma \in (0,1]$. Then, the generalized Burr distribution is a scale mixture of the uniform distribution:

$$|T_{r,\alpha }{|}^{1/\gamma }\stackrel{d}{=}{Y}_{r,\alpha ,\gamma }\circ G_{2,1}\circ R_{[0,1]}.$$

(32)

where the random variable $Y_{r,\alpha ,\gamma }$ was defined in Proposition 1.

Proof. 

It suffices to notice that Lemma 4 with $\alpha =1$ implies $|Q_1|\stackrel{d}{=}{W}_1\stackrel{d}{=}{R}_{[0,1]}\circ G_{2,1}$. □

By virtue of (26), Proposition 5 implies the following result.

Corollary 14.

Let $r_1\ge 1$ and $r_2>r_1$. Then, the Snedecor–Fisher distribution is a scale mixture of the uniform distribution:

$$P_{r_1,r_2}\stackrel{d}{=}{Y}_{r_2,1/r_1,r_1}\circ G_{2,r_1}\circ R_{[0,1]}.$$

(33)

where the random variable $Y_{r_2,1/r_1,r_1}$ was defined in Proposition 1.

4. Asymmetric Generalized Student and Two-Sided Generalized Burr Distributions

4.1. Randomized Asymmetrization of the Generalized Student and Two-Sided Generalized Burr Distributions

For $r>0$ and $\alpha \in (0,2]$, let $F_{r,\alpha }\left(x\right)=\mathsf{P}(T_{r,\alpha }<x)$, $x\in \mathbb{R}$. Then

$$\mathsf{P}\left(\right|T_{r,\alpha }|<x)=2F_{r,\alpha }\left(x\right)-1,x\ge 0.$$

As the starting point for the construction of the randomized de-symmetrization of the generalized Student distribution, we will take the following representation of the generalized Student distribution as a mixture of the discrete uniform (Bernoulli) distributions with the Snedecor–Fisher mixing distribution.

For $p\in (0,1)$, let $I_p$ be the Bernoulli random variable: $\mathsf{P}(I_p=1)=p=1-\mathsf{P}(I_p=0)$. Then, the random variable $2I_p-1$ takes two values 1 and $-1$ with probabilities p and $1-p$.

Since the generalized Student distribution is symmetric, from relation (26), it follows that

$${\alpha }^{1/\alpha }{T}_{r,\alpha }\stackrel{d}{=}(2I_{1/2}-1)\circ P_{1/\alpha ,r}^{1/\alpha }.$$

(34)

Along with (34), another Bernoulli mixture representation is possible for $T_{r,\alpha }$:

$${\alpha }^{1/\alpha }{T}_{r,\alpha }\stackrel{d}{=}{I}_{1/2}\circ P_{1/\alpha ,r}^{1/\alpha }-(1-I_{1/2})\circ P_{1/\alpha ,r}^{1/\alpha }\stackrel{d}{=}{\left[I_{1/2}\circ P_{1/\alpha ,r}-(1-I_{1/2})\circ P_{1/\alpha ,r}\right]}^{1/\alpha }.$$

(35)

To construct the ‘randomized’ asymmetrization of the generalized Student distribution, we will take different multipliers at the Bernoulli random variables.

For $i=1,2$, let $r_i>0$, ${\alpha }_i\in (0,2]$. Let $T_{r_1,{\alpha }_1}$ and $T_{r_2,{\alpha }_2}$ be two random variables independent of I. Define the random variable $T_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}$ as

$${\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}=I_p\circ |T_{r_1,{\alpha }_1}|-(1-I_p)\circ \left|T_{r_2,{\alpha }_2}\right|.$$

The distribution of the random variable $T_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}$ will be called R-asymmetric (randomization-asymmetric) generalized Student distribution.

This distribution, for example, can be used as a heavy-tailed model for the distribution of an elementary increment of a process describing the evolution of a stock price or a financial index: the random variable $I_p$ indicates the increase or decrease in the process (that is, the sign of the increment), whereas the generalized folded Student distributions describe the statistical regularities of the absolute values of positive and negative increments.

Denote ${\overline{F}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)=\mathsf{P}({\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}<x)$. From the definition of the random variable ${\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}$, it follows that the distribution function ${\overline{F}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)$ is a mixture of two distribution functions:

$${\overline{F}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)=p\mathsf{P}\left(\right|T_{r_1,{\alpha }_1}|<x)+(1-p)\mathsf{P}(-|T_{r_2,{\alpha }_2}|<x).$$

Moreover, it is not difficult to verify that

$${\overline{F}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)=\left\{\begin{array}{cc}2(1-p)F_{r_2,{\alpha }_2}\left(x\right),\hfill & x\le 0;\hfill \\ 1+2p(F_{r_1,{\alpha }_1}\left(x\right)-1),\hfill & x>0,\hfill \end{array}\right.$$

so that $\mathsf{P}({\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\le 0)=1-p$.

Consequently, the corresponding probability density function ${\overline{f}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)$ has the form

$${\overline{f}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)=\left\{\begin{array}{cc}2(1-p)f_{r_2,{\alpha }_2}\left(x\right),\hfill & x<0,\hfill \\ 2p{f}_{r_1,{\alpha }_1}\left(x\right),\hfill & x>0.\hfill \end{array}\right.$$

As this is so, ${\overline{f}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(0\right)$ can be defined either as $2(1-p)f_{r_2,{\alpha }_2}\left(0\right)$, or as $2p{f}_{r_1,{\alpha }_1}\left(0\right)$. However, in general, it is possible that ${\overline{f}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}\left(x\right)$ can be discontinuous at the point $x=0$. To avoid this, the additional condition has to be assumed:

$$(1-p)f_{r_2,{\alpha }_2}\left(0\right)=p{f}_{r_1,{\alpha }_1}\left(0\right),$$

or with the account of (17),

$$\frac{{\alpha }_1{r}_2^{1/{\alpha }_2}\mathrm{B}(r_2,\frac{1}{{\alpha }_2})}{{\alpha }_2{r}_1^{1/{\alpha }_1}\mathrm{B}(r_1,\frac{1}{{\alpha }_1})}=\frac{1-p}{p}.$$

This condition formally reduces the number of the parameters of the R-asymmetric generalized Student distribution from five to four. However, in order to meet this condition, a special additional auxiliary parameter has to be introduced, for example, in the way it was carried out in [29,30].

The R-asymmetric generalized Student distribution is obviously unimodal. The form of the vertex of the R-asymmetric generalized Student distribution density depends on the values of ${\alpha }_1$ and ${\alpha }_2$. If both of these parameters are no less than 1, then the vertex is flat. If at least one of ${\alpha }_1$ or ${\alpha }_2$ is less than 1, then the vertex has the form of an angle; moreover, if only one of these parameters is less than one, then this angle is nonzero, and if both parameters are less than 1, then the vertex is ‘infinitely sharp’, i.e., the angle is zero.

The left and right tails if the R-asymmetric generalized Student distribution have power type of decrease, possibly with different exponents: if $x\to \infty$, then $\mathsf{P}({\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}<-x)=O\left(x^{-{\alpha }_2{r}_2}\right)$ and $\mathsf{P}({\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}>x)=O\left(x^{-{\alpha }_1{r}_1}\right)$.

It is not difficult to verify that for any $\delta \in (-1,\alpha r)$

$$\mathsf{E}|T_{r,\alpha }{|}^{\delta }=\mathsf{E}{G}_{r,r}^{-\delta /\alpha }·\mathsf{E}{\left|Q_{\alpha }\right|}^{\delta }=\frac{r^{\delta /\alpha }\mathsf{\Gamma }(r-\frac{\delta }{\alpha })\mathsf{\Gamma }\left(\frac{\delta +1}{\alpha }\right)}{\mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)}$$

(36)

(see Proposition 2 in [1]). Therefore, if $\min \{{\alpha }_1{r}_2,{\alpha }_2{r}_2\}>1$, then the expectation of the random variable with the R-asymmetric generalized Student distribution has the form

$$\mathsf{E}{\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}=p\mathsf{E}|T_{r_1,{\alpha }_1}|-(1-p)\mathsf{E}|T_{r_2,{\alpha }_2}|=$$

$$=\frac{p{r}_1^{1/{\alpha }_1}\mathsf{\Gamma }(r_1-\frac{1}{{\alpha }_1})\mathsf{\Gamma }\left(\frac{2}{{\alpha }_1}\right)}{\mathsf{\Gamma }\left(r_1\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_1}\right)}-\frac{(1-p)r_1^{1/{\alpha }_2}\mathsf{\Gamma }(r_2-\frac{1}{{\alpha }_2})\mathsf{\Gamma }\left(\frac{2}{{\alpha }_2}\right)}{\mathsf{\Gamma }\left(r_2\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_2}\right)};$$

Moreover, if $\min \{{\alpha }_1{r}_2,{\alpha }_2{r}_2\}>2$, then

$$\mathsf{E}{\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}^2=p\mathsf{E}{T}_{r_1,{\alpha }_1}^2+(1-p)\mathsf{E}{T}_{r_2,{\alpha }_2}^2=$$

$$=\frac{p{r}_1^{2/{\alpha }_1}\mathsf{\Gamma }(r_1-\frac{2}{{\alpha }_1})\mathsf{\Gamma }\left(\frac{3}{{\alpha }_1}\right)}{\mathsf{\Gamma }\left(r_1\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_1}\right)}+\frac{(1-p)r_1^{2/{\alpha }_2}\mathsf{\Gamma }(r_2-\frac{2}{{\alpha }_2})\mathsf{\Gamma }\left(\frac{3}{{\alpha }_2}\right)}{\mathsf{\Gamma }\left(r_2\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_2}\right)},$$

so that

$$\mathsf{D}{\overline{T}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}=\frac{p{r}_1^{2/{\alpha }_1}\mathsf{\Gamma }(r_1-\frac{2}{{\alpha }_1})\mathsf{\Gamma }\left(\frac{3}{{\alpha }_1}\right)}{\mathsf{\Gamma }\left(r_1\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_1}\right)}+\frac{(1-p)r_1^{2/{\alpha }_2}\mathsf{\Gamma }(r_2-\frac{2}{{\alpha }_2})\mathsf{\Gamma }\left(\frac{3}{{\alpha }_2}\right)}{\mathsf{\Gamma }\left(r_2\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_2}\right)}-$$

$$-{\left(\frac{p{r}_1^{1/{\alpha }_1}\mathsf{\Gamma }(r_1-\frac{1}{{\alpha }_1})\mathsf{\Gamma }\left(\frac{2}{{\alpha }_1}\right)}{\mathsf{\Gamma }\left(r_1\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_1}\right)}-\frac{(1-p)r_1^{1/{\alpha }_2}\mathsf{\Gamma }(r_2-\frac{1}{{\alpha }_2})\mathsf{\Gamma }\left(\frac{2}{{\alpha }_2}\right)}{\mathsf{\Gamma }\left(r_2\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_2}\right)}\right)}^2=$$

$$=p\mathsf{D}|T_{r_1,{\alpha }_1}|+(1-p)\mathsf{D}|T_{r_2,{\alpha }_2}|+2p(1-p)·\mathsf{E}|T_{r_1,{\alpha }_1}|·\mathsf{E}|T_{r_2,{\alpha }_2}|.$$

Similarly, for any $n\in \mathbb{N}$, $n<\min \{{\alpha }_1{r}_1,{\alpha }_2{r}_2\}$, we have

$$\mathsf{E}{T}_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}^n=\frac{p{r}_1^{n/{\alpha }_1}\mathsf{\Gamma }(r_1-\frac{n}{{\alpha }_1})\mathsf{\Gamma }\left(\frac{n+1}{{\alpha }_1}\right)}{\mathsf{\Gamma }\left(r_1\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_1}\right)}+{(-1)}^n\frac{(1-p)r_2^{n/{\alpha }_2}\mathsf{\Gamma }(r_2-\frac{n}{{\alpha }_2})\mathsf{\Gamma }\left(\frac{n+1}{{\alpha }_2}\right)}{\mathsf{\Gamma }\left(r_2\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_2}\right)},$$

(37)

and for any $\delta \in (-1,\min \{{\alpha }_1{r}_1,{\alpha }_2{r}_2\})$

$$\mathsf{E}|T_{r_1,r_2,{\alpha }_1,{\alpha }_2,p}{|}^{\delta }=\frac{p{r}_1^{\delta /{\alpha }_1}\mathsf{\Gamma }(r_1-\frac{\delta }{{\alpha }_1})\mathsf{\Gamma }\left(\frac{\delta +1}{{\alpha }_1}\right)}{\mathsf{\Gamma }\left(r_1\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_1}\right)}+\frac{(1-p)r_2^{\delta /{\alpha }_2}\mathsf{\Gamma }(r_2-\frac{\delta }{{\alpha }_2})\mathsf{\Gamma }\left(\frac{\delta +1}{{\alpha }_2}\right)}{\mathsf{\Gamma }\left(r_2\right)\mathsf{\Gamma }\left(\frac{1}{{\alpha }_2}\right)}.$$

(38)

As concerns the two-sided generalization of the Burr distribution, first, by analogy with (34), for $r>0$, $\alpha \in (0,2]$ and $\gamma >0$, define the symmetric two-sided generalized Burr distribution as the distribution of the random variable ${\overline{B}}_{r,\alpha ,\gamma }$ such that

$${\alpha }^{1/\left(\alpha \gamma \right)}{\overline{B}}_{r,\alpha ,\gamma }\stackrel{d}{=}(2I_{1/2}-1)\circ P_{1/\alpha ,r}^{1/\left(\alpha \gamma \right)}.$$

(39)

Then, following the general randomization scheme described above, consider $p\in [0,1]$, $r_i>0$, ${\alpha }_i\in (0,2]$, ${\gamma }_i>0$, $i=1,2$. Define the R-asymmetric two-sided generalized Burr distribution as the distribution of the random variable

$${\overline{B}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,\gamma ,p}=I_p|T_{r_1,{\alpha }_1}{|}^{1/{\gamma }_1}-(1-I_p){\left|T_{r_2,{\alpha }_2}\right|}^{1/{\gamma }_2}\stackrel{d}{=}$$

$$\stackrel{d}{=}{\alpha }_1^{-1/\left({\alpha }_1{\gamma }_1\right)}{I}_p\circ P_{1/{\alpha }_1,r_1}^{1/\left({\alpha }_1{\gamma }_1\right)}-{\alpha }_2^{-1/\left({\alpha }_2{\gamma }_2\right)}(1-I_p)\circ P_{1/{\alpha }_2,r_2}^{1/\left({\alpha }_2{\gamma }_2\right)}.$$

Correspondingly, the density ${\overline{b}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,\gamma ,p}\left(x\right)$ of the R-asymmetric two-sided generalized Burr distribution has the form

$${\overline{b}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,\gamma ,p}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{p{\alpha }_1{\gamma }_1{x}^{{\gamma }_1-1}}{r_1^{1/{\alpha }_1}\mathrm{B}(r_1,\frac{1}{{\alpha }_1})}{\left(1+\frac{x^{{\alpha }_1{\gamma }_1}}{r_1}\right)}^{-(r_1+1/{\alpha }_1)},}\hfill & x\ge 0,\hfill \\ {\displaystyle \frac{(1-p){\alpha }_2{\gamma }_2{\left|x\right|}^{{\gamma }_2-1}}{r_2^{1/{\alpha }_2}\mathrm{B}(r_2,\frac{1}{{\alpha }_2})}{\left(1+\frac{{\left|x\right|}^{{\alpha }_2{\gamma }_2}}{r_2}\right)}^{-(r_2+1/{\alpha }_2)},}\hfill & x<0.\hfill \end{array}\right.$$

(40)

If at least one of ${\gamma }_1$ or ${\gamma }_2$ exceeds 1, then this density is bimodal. If both of these parameters do not exceed 1, then the density is unimodal with mode in zero. In the former case, the vertices of the density are finite only if both of ${\gamma }_1$ and ${\gamma }_2$ are greater than 1. In the latter case, the vertex of ${\overline{b}}_{r_1,r_2,{\alpha }_1,{\alpha }_2,\gamma ,p}\left(x\right)$ is finite only if ${\gamma }_1={\gamma }_2=1$.

In exactly the same way as the two-sided generalized Burr distribution was defined, by virtue of (26), we can define the symmetric and R-asymmetric two-sided Snedecor–Fisher distribution. The expressions for the moments of this distribution can be easily obtained from those for the moments of the R-asymmetric two-sided generalized Burr distribution by a simple re-denotation of the parameters.

4.2. Asymmetrization of the Generalized Student and Two-Sided Generalized Burr Distributions by Normal Variance-Mean Mixing

Another approach to the asymmetrization of the generalized Student and two-sided generalized Burr distributions is based on normal mixture representations of these distributions. The starting points here are Lemma 9 and Definition 1, respectively. We begin with generalized Student distribution. According to Lemma 9, the symmetric generalized Student distribution is a normal scale mixture:

$$T_{r,\alpha }\stackrel{d}{=}\sqrt{D_{r,\alpha }}\circ X,$$

where

$$D_{r,\alpha }\stackrel{d}{=}\frac{1}{2}{U}_{2,\alpha /2}^{-1}\circ G_{r,r}^{-2/\alpha }.$$

For the properties of the random variable $D_{r,\alpha }$ see [1].

Following the general lines of the construction of normal variance–mean mixtures [31], we introduce two scalar parameters, scale parameter $\sigma >0$ and asymmetry parameter $a\in \mathbb{R}$, and define the M-asymmetric (mixture-asymmetric) generalized Student distribution function ${\tilde{F}}_{r,\alpha ,a,b}\left(x\right)$ as

$${\tilde{F}}_{r,\alpha ,a,\sigma }\left(x\right)={\int }_0^{\infty }\mathsf{\Phi }\left(\frac{x-au}{\sigma \sqrt{u}}\right)d\mathsf{P}(D_{r,\alpha }<u),x\in \mathbb{R}.$$

(41)

This distribution function corresponds to the random variable

$${\tilde{T}}_{r,\alpha ,a,\sigma }=\sigma \sqrt{D_{r,\alpha }}\circ X+a{D}_{r,\alpha }.$$

We find the moments of the random variable ${\tilde{T}}_{r,\alpha ,a,\sigma }$. In [13], it was shown that if $\delta \ge -\frac{1}{\alpha }$, then

$$\mathsf{E}{U}_{\alpha ,{\alpha }^{\prime }}^{-\delta }=\frac{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{\delta \alpha +1}{\alpha {\alpha }^{\prime }}\right)}{\mathsf{\Gamma }\left(\frac{1}{\alpha {\alpha }^{\prime }}\right)\mathsf{\Gamma }\left(\frac{\delta \alpha +1}{\alpha }\right)}.$$

(42)

It is well known that if $\delta >r$, then

$$\mathsf{E}{G}_{r,r}^{-\delta }=\frac{r^r}{\mathsf{\Gamma }\left(r\right)}{\int }_0^{\infty }{x}^{r-\delta -1}{e}^{-rx}dx=\frac{r^{\delta }\mathsf{\Gamma }(r-\delta )}{\mathsf{\Gamma }\left(r\right)}$$

Hence, if $-\frac{1}{2}<\delta <\frac{\alpha r}{2}$, then

$$\mathsf{E}{D}_{r,\alpha }^{\delta }=\frac{1}{2^{\delta }}\mathsf{E}{U}_{2,\alpha /2}^{-\delta }·\mathsf{E}{G}_{r,r}^{-2\delta /\alpha }=\frac{\sqrt{\pi }}{2^{\delta }}·\frac{\mathsf{\Gamma }\left(\frac{2\delta +1}{\alpha }\right)}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }(\delta +\frac{1}{2})}·\frac{r^{2\delta /\alpha }\mathsf{\Gamma }(r-\frac{2\delta }{\alpha })}{\mathsf{\Gamma }\left(r\right)}.$$

So, if $\alpha r>2$, then

$$\mathsf{E}{\tilde{T}}_{r,\alpha ,a,\sigma }=a\mathsf{E}{D}_{r,\alpha }=\frac{a{r}^{2/\alpha }\mathsf{\Gamma }\left(\frac{3}{\alpha }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha })}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}.$$

If $\alpha r>4$, then

$$\mathsf{E}{\tilde{T}}_{r,\alpha ,a,\sigma }^2={\sigma }^2\mathsf{E}{D}_{r,\alpha }+a^2\mathsf{E}{D}_{r,\alpha }^2=\frac{{\sigma }^2{r}^{2/\alpha }\mathsf{\Gamma }\left(\frac{3}{\alpha }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha })}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}+\frac{a^2{r}^{4/\alpha }\mathsf{\Gamma }\left(\frac{5}{\alpha }\right)\mathsf{\Gamma }(r-\frac{4}{\alpha })}{3\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}.$$

Therefore,

$$\mathsf{D}{\tilde{T}}_{r,\alpha ,a,\sigma }=\frac{r^{2/\alpha }\mathsf{\Gamma }\left(\frac{3}{\alpha }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha })}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}\left({\sigma }^2-\frac{a^2{r}^{2/\alpha }\mathsf{\Gamma }\left(\frac{3}{\alpha }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha })}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}\right)+\frac{a^2{r}^{4/\alpha }\mathsf{\Gamma }\left(\frac{5}{\alpha }\right)\mathsf{\Gamma }(r-\frac{4}{\alpha })}{3\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}.$$

Consider the higher-order moments of the M-asymmetric generalized Student distribution. Let $n\in \mathbb{N}$, $n\ge 3$. It is not difficult to verify that if $\alpha r>n$, then

$$\mathsf{E}{\tilde{T}}_{r,\alpha ,a,\sigma }^n=\frac{n!\sqrt{\pi }}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(r\right)}\sum _{j=0}^{[n/2]}\frac{{\sigma }^{2j}(2j-1)!!\mathsf{\Gamma }\left(\frac{2(n-j)+1}{\alpha }\right)\mathsf{\Gamma }(r-\frac{2(n-j)}{\alpha })}{\left(2j\right)!(n-2j)!\mathsf{\Gamma }(n-j+\frac{1}{2})}{\left(\frac{a{r}^{2/\alpha }}{2}\right)}^{n-j}.$$

As regards the tail behavior of the M-asymmetric generalized Student distribution, we will require the following simple auxiliary general result.

Lemma 11.

Let Z and Y be two random variables. Then, for any $q\in (0,1)$ and $x>0$

$$\max \left\{\mathsf{P}\left(\right|Z|\ge x),\mathsf{P}\left(\right|Y|\ge x)\right\}\le \mathsf{P}\left(\right|Z+Y|\ge x)\le \mathsf{P}\left(\left|Z\right|\ge qx\right)+\mathsf{P}\left(\left|Y\right|\ge (1-q)x\right).$$

Proof. 

This assertion follows from obvious implications

$$\left(\{\omega :|Z\left(\omega \right)|\ge x\}\cup \{\omega :|Y\left(\omega \right)|\ge x\}\right)\subseteq \{\omega :|Z\left(\omega \right)+Y\left(\omega \right)|\ge x\}\subseteq$$

$$\subseteq \left(\{\omega :|Z\left(\omega \right)|\ge qx\}\cup \{\omega :|Y\left(\omega \right)|\ge (1-q\left)x\right\}\right).$$

□

In [1], it was shown that $\mathsf{P}\left(\right|T_{r,\alpha }|\ge x)=O\left(x^{-r\alpha }\right)$ and $\mathsf{P}(D_{r,\alpha }\ge x)=O\left(x^{-r\alpha /2}\right)$ as $x\to \infty$. So, based on Lemma 11, we can conclude that as $x\to \infty$,

$$\mathsf{P}\left(\right|{\tilde{T}}_{r,\alpha ,a,\sigma }|\ge x)=O\left(x^{-r\alpha /2}\right).$$

Now, we turn to the M-asymmetrization of the two-sided generalized Burr distribution. Since the random variable with the generalized Student distribution takes values of both signs, in general, it is impossible to correctly define the power of this random variable with an arbitrary real exponent. The formal ‘direct’ definitions are correct only for non-negative integer powers. However, the statement of Corollary 7 gives grounds for the formally correct ‘indirect’ definition of the two-sided generalized Burr distribution (or, which is the same, the quasi-exponentiated generalized Student distribution) for an arbitrary real exponent as a special normal scale mixture.

Definition 1.

Let $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma \in (0,1]$. The distribution of the random variable $\sqrt{Y_{r,\alpha ,\gamma }^{*}}\circ X$, where the random variable $Y_{r,\alpha ,\gamma }^{*}$ was defined in Corollary 7, is called the quasi-exponentiated generalized Student distribution or the two-sided generalized Burr distribution.

This definition opens the way for the construction of the asymmetric quasi-exponentiated generalized Student distributions (or the asymmetric two-sided generalized Burr distributions, which are the same) that will be considered below.

It should be especially noted that unlike the norm in probability statistics, where the exponentiated distribution function is meant by ‘exponentiated distribution’, in Definition 1, by this term we mean the distribution of the (quasi-) exponentiated random variable.

From Corollary 7, we obviously have $\sqrt{Y_{r,\alpha ,\gamma }^{*}}\circ X\stackrel{d}{=}{T}_{r,\alpha ,\gamma }\stackrel{d}{=}(2I_{1/2}-1)\circ {\left|T_{r,\alpha }\right|}^{1/\gamma }\stackrel{\mathrm{def}}{=}{\overline{B}}_{r,\alpha ,\gamma }$, where the random variables $I_{1/2}$ and $T_{r,\alpha }$ are independent. Moreover, if $1/\gamma$ is an odd number, then ${\overline{B}}_{r,\alpha ,\gamma }\stackrel{d}{=}{T}_{r,\alpha }^{1/\gamma }$.

Proposition 6.

Let $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma \in (0,1]$. Then, the two-sided generalized Burr distribution is infinitely divisible.

Proof. 

Recall that the random variable $Y_{r,\alpha ,\gamma }^{*}$ is represented as $Y_{r,\alpha ,\gamma }^{*}=2Y_{r,\alpha ,\gamma }^2\circ W_1$, with the random variable $Y_{r,\alpha ,\gamma }$ defined in Proposition 1. According to the result of [23] mentioned above, the distribution of $Y_{r,\alpha ,\gamma }^{*}$ is infinitely divisible. In accordance with [32], Ch. XVII, Sect. 3, the normal scale mixture is infinitely divisible if the mixing distribution is infinitely divisible, from which follows the desired assertion. □

Now, again following the general lines of the construction of normal variance–mean mixtures, introduce two scalar parameters: scale parameter $\sigma >0$ and asymmetry parameter $a\in \mathbb{R}$, and define the M-asymmetric two-sided generalized Burr distribution function ${\tilde{A}}_{r,\alpha ,\gamma ,a,\sigma }\left(x\right)$ as

$${\tilde{A}}_{r,\alpha ,\gamma ,a,\sigma }\left(x\right)={\int }_0^{\infty }\mathsf{\Phi }\left(\frac{x-au}{\sigma \sqrt{u}}\right)d\mathsf{P}(Y_{r,\alpha ,\gamma }^{*}<u),x\in \mathbb{R}.$$

(43)

This distribution function corresponds to the random variable

$${\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }=\sigma \sqrt{Y_{r,\alpha ,\gamma }^{*}}\circ X+a{Y}_{r,\alpha ,\gamma }^{*}.$$

Find the moments of ${\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }$. Let $\alpha \in (0,1]$, $\gamma \in (0,1]$ and $\alpha \gamma r>2$. We have

$$\mathsf{E}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }=a\mathsf{E}{Y}_{r,\alpha ,\gamma }^{*}=a\mathsf{E}{U}_{1,\alpha }^{-2/\gamma }·\mathsf{E}{G}_{r,r}^{-2/\left(\alpha \gamma \right)}·\mathsf{E}{S}_{\gamma ,1}^{-2}·\mathsf{E}{W}_1.$$

(44)

In accordance with (42), we have

$$\mathsf{E}{U}_{1,\alpha }^{-2/\gamma }=\frac{\mathsf{\Gamma }\left(\frac{2+\gamma }{\alpha \gamma }\right)}{\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\gamma }\right)}.$$

(45)

Further,

$$\mathsf{E}{G}_{r,r}^{-2/\left(\alpha \gamma \right)}=\frac{r^{2/\left(\alpha \gamma \right)}\mathsf{\Gamma }(r-\frac{2}{\alpha \gamma })}{\mathsf{\Gamma }\left(r\right)}.$$

(46)

According to Lemma 1,

$$\mathsf{E}{S}_{\gamma ,1}^{-2}=\frac{1}{\gamma }\mathsf{\Gamma }\left(\frac{2}{\gamma }\right).$$

(47)

Since $\mathsf{E}{W}_1=1$, from (44)–(47) we obtain

$$\mathsf{E}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }=\frac{a{r}^{2/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{2}{\gamma }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha \gamma })}{\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\gamma }\right)}.$$

Now let $\alpha \in (0,1]$, $\gamma \in (0,1]$ and $\alpha \gamma r>4$. Using (42) and (5) with the account of the relations $\mathsf{E}{W}_1^2=2$ and

$$\mathsf{E}{G}_{r,r}^{-4/\left(\alpha \gamma \right)}=\frac{r^{4/\left(\alpha \gamma \right)}\mathsf{\Gamma }(r-\frac{4}{\alpha \gamma })}{\mathsf{\Gamma }\left(r\right)},$$

we obtain

$$\mathsf{E}{\left(Y_{r,\alpha ,\gamma }^{*}\right)}^2=\mathsf{E}{U}_{1,\alpha }^{-4/\gamma }·\mathsf{E}{G}_{r,r}^{-4/\left(\alpha \gamma \right)}·\mathsf{E}{S}_{\gamma ,1}^{-4}·\mathsf{E}{W}_1^2=\frac{r^{4/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{4}{\gamma }\right)\mathsf{\Gamma }\left(\frac{4+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{4}{\alpha \gamma })}{3\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{4+\gamma }{\gamma }\right)}.$$

Hence,

$$\mathsf{E}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }^2={\sigma }^2\mathsf{E}{Y}_{r,\alpha ,\gamma }^{*}+a^2\mathsf{E}{\left(Y_{r,\alpha ,\gamma }^{*}\right)}^2=$$

$$=\frac{{\sigma }^2{r}^{2/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{2}{\gamma }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha \gamma })}{\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\gamma }\right)}+\frac{a^2{r}^{4/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{4}{\gamma }\right)\mathsf{\Gamma }\left(\frac{4+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{4}{\alpha \gamma })}{3\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{4+\gamma }{\gamma }\right)}.$$

Therefore,

$$\mathsf{D}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }=\mathsf{E}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }^2-{\left(\mathsf{E}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }\right)}^2=$$

$$=\frac{r^{2/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{2}{\gamma }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha \gamma })}{\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\gamma }\right)}\left[{\sigma }^2-\frac{a^2{r}^{2/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{2}{\gamma }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{2}{\alpha \gamma })}{\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{2+\gamma }{\gamma }\right)}\right]+$$

$$+\frac{a^2{r}^{4/\left(\alpha \gamma \right)}\mathsf{\Gamma }\left(\frac{4}{\gamma }\right)\mathsf{\Gamma }\left(\frac{4+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{4}{\alpha \gamma })}{3\gamma \mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }\left(\frac{4+\gamma }{\gamma }\right)}.$$

The moments of higher orders can be calculated by the binomial formula. Let $n\in \mathbb{N}$, $3\le n<\frac{\alpha \gamma r}{2}$. Then, using (42), (5) and well-known expressions for the moments of the gamma distribution, we obtain

$$\mathsf{E}{\tilde{B}}_{r,\alpha ,\gamma ,a,\sigma }^n=\mathsf{E}{\left(\sigma \sqrt{Y_{r,\alpha ,\gamma }^{*}}\circ X+a{Y}_{r,\alpha ,\gamma }^{*}\right)}^n=\mathsf{E}\sum _{j=0}^n{C}_n^j{\sigma }^j{a}^{n-j}{X}^j\circ {\left(Y_{r,\alpha ,\gamma }^{*}\right)}^{n-j/2}=$$

$$=\sum _{j=0}^{[n/2]}{C}_n^{2j}{\sigma }^{2j}{a}^{n-2j}\mathsf{E}{X}^{2j}\mathsf{E}{\left(Y_{r,\alpha ,\gamma }^{*}\right)}^{n-j}=$$

$$=\frac{1}{\sqrt{\pi }}\sum _{j=0}^{[n/2]}{C}_n^{2j}{2}^j{r}^{2(n-j)/\left(\alpha \gamma \right)}{\sigma }^{2j}{a}^{n-2j}\mathsf{\Gamma }(j+\frac{1}{2})\mathsf{\Gamma }(n-j+1)\mathsf{E}{U}_{1,\alpha }^{-2(n-j)/\gamma }\mathsf{E}{G}_{r,1}^{-2(n-j)/\left(\alpha \gamma \right)}\mathsf{E}{S}_{\gamma ,1}^{-2(n-j)}=$$

$$=\frac{1}{\sqrt{\pi }}\sum _{j=0}^{[n/2]}{C}_n^{2j}{2}^j{r}^{2(n-j)/\left(\alpha \gamma \right)}{\sigma }^{2j}{a}^{n-2j}\frac{\mathsf{\Gamma }(j+\frac{1}{2})\mathsf{\Gamma }(n-j+1)\mathsf{\Gamma }\left(\frac{2(n-j)+\gamma }{\alpha \gamma }\right)\mathsf{\Gamma }(r-\frac{2(n-j)}{\alpha \gamma })}{\mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(\frac{1}{\alpha }\right)\mathsf{\Gamma }(2(n-j)+1)}.$$

In exactly the same way as the R-asymmetric two-sided generalized Burr distribution was defined, by virtue of (26), we can define the M-asymmetric two-sided Snedecor–Fisher distribution. The expressions for the moments of this distribution can be easily obtained from those for the moments of the M-asymmetric two-sided generalized Burr distribution by a simple re-denotation of the parameters.

5. Generalized Burr, M-Asymmetric Two-Sided Generalized Burr and M-Asymmetric Generalized Student Distributions as Limit Laws in Limit Theorems for Statistics Constructed from Samples with Random Sizes

In this section, we will demonstrate that the generalized Burr, symmetric and M-asymmetric two-sided generalized Burr and M-asymmetric generalized Student distributions can be limit laws in rather simple limit theorems with a rather general structure for some conventional statistics constructed from samples with random sizes. These theorems can be used both as motivation and theoretical grounds for the use of these (in general, heavy-tailed) models as asymptotic approximations in practical problems. As examples of such theorems, we will present limit theorems for random sums in which the limit laws are the M-asymmetric generalized Student distribution as well as the symmetric and M-asymmetric two-sided generalized Burr distribution, and also limit theorems for maximum random sums and extreme order statistics, in which the limit law is the generalized Burr distribution.

5.1. Limit Theorem for Random Sums with Generalized Burr Limit Distribution

The representation of the generalized Burr as a scale mixture of the folded normal distribution (see Corollary 7) makes it possible to formulate a limit theorem for maximum random sums (that is, for the maximum value attained by a randomly stopped random walk). More exactly, we will demonstrate that the generalized Burr distribution can be the limit law for maximum sums of a random number of independent random variables (maximum random sums), minimum random sums and absolute values of random sums and, at the same time, the symmetric two-sided generalized Burr distribution can be the limit law for random sums themselves.

Let $X_1,X_2,\dots$ be independent not necessarily identically distributed random variables with $\mathsf{E}{X}_i=0$ and $0<{\sigma }_i^2=\mathsf{E}{X}_i^2<\infty$, $i\ge 1$. For $n\in \mathbb{N}$, denote

$$S_n=X_1+\dots +X_n,{\sum }_n^2={\sigma }_1^2+\dots +{\sigma }_n^2.$$

Assume that the random variables $X_1,X_2,\dots$ satisfy the Lindeberg condition: for any $\tau >0$

$$\underset{n\to \infty }{\lim }\frac{1}{{\sum }_n^2}\sum _{i=1}^n{\int }_{\left|x\right|\ge \tau {\sum }_n}^{}{x}^{2}d\mathsf{P}(X_i<x)=0.$$

(48)

The well-known Lindeberg central limit theorem states that under these assumptions

$$\mathsf{P}\left(S_n<{\sum }_{n}x\right)⟹\mathsf{\Phi }\left(x\right).$$

(49)

Let $N_1,N_2,\dots$ be a non-negative integer-valued random variables such that for each $n\in \mathbb{N}$, the random variables $N_n,X_1,X_2,\dots$ are independent. Denote $S_{N_n}=X_1+\dots +X_{N_n}$. For definiteness, in what follows we will assume that ${\sum }_{j=1}^0=0$ and the convergence is meant as $n\to \infty$.

Recall that a random sequence $N_1,N_2,\dots$ is called, infinitely increasing in probability if $\mathsf{P}(N_n\le m)⟶0$ for any $m\in (0,\infty )$.

For $n\in \mathbb{N}$, denote ${\overline{S}}_n={\max }_{1\le i\le n}{S}_i$, ${\underline{S}}_n={\min }_{1\le i\le n}{S}_i$. It is well known that under condition (48) along with (49) we also have

$$\mathsf{P}\left({\overline{S}}_n<{\sum }_{n}x\right)⟹2\mathsf{\Phi }\left(x\right)-1,x\ge 0,$$

and

$$\mathsf{P}\left({\underline{S}}_n<{\sum }_{n}x\right)⟹2\mathsf{\Phi }\left(x\right),x\le 0.$$

For $n\in \mathbb{N}$ let $S_{N_n}=X_1+\dots +X_{N_n}$, ${\overline{S}}_{N_n}={\max }_{1\le i\le N_n}{S}_i$, ${\underline{S}}_{N_n}={\min }_{1\le i\le N_n}{S}_i$ (for definiteness, assume that $S_0={\overline{S}}_0={\underline{S}}_0=0$). Let ${\left\{d_n\right\}}_{n\ge 1}$ be an arbitrary infinitely increasing sequence of positive numbers.

Lemma 12.

[33]. Assume that the random variables $X_1,X_2,\dots$ and $N_1,N_2,\dots$ satisfy the conditions specified above. In particular, let Lindeberg condition (48) hold and $N_n\to \infty$ in probability. Then, the distributions of normalized random sums converge to some distribution, i.e., there exists a random variable V such that

$$d_n^{-1}{S}_{N_n}⟹V,$$

if and only if there exists a non-negative random variable Y such that $V\stackrel{d}{=}\sqrt{Y}\circ X$ and any of the following conditions holds:

(i) 

$d_n^{-1}|S_{N_n}|⟹|V|$;

(ii) 

There exists a random variable $\overline{V}$ such that $d_n^{-1}{\overline{S}}_{N_n}⟹\overline{V}$;

(iii) 

There exists a random variable $\underline{V}$ such that $d_n^{-1}{\underline{S}}_{N_n}⟹\underline{V}$;

(iv) 

There exists a non-negative random variable Y such that $d_n^{-2}{\sum }_{N_n}^2⟹Y$.

Moreover,

$$\mathsf{P}\left(\underline{V}<x\right)=2\mathsf{E}\mathsf{\Phi }\left(x{Y}^{-1/2}\right),x\le 0;\mathsf{P}\left(\overline{V}<x\right)=\mathsf{P}\left(\left|V\right|<x\right)=2\mathsf{E}\mathsf{\Phi }\left(x{Y}^{-1/2}\right)-1,x\ge 0.$$

Lemma 12 and Corollary 7 imply the following statement.

Proposition 7.

Let $\alpha \in (0,1]$, $\alpha r>1$ and $\gamma \in (0,1]$. Assume that the random variables $X_1,X_2,\dots$ and $N_1,N_2,\dots$ satisfy the conditions specified above. In particular, let Lindeberg condition (48) hold. Moreover, let $N_n\to \infty$ in probability. Then, the following five statements are equivalent:

$$d_n^{-1}{S}_{N_n}⟹{\overline{B}}_{r,\alpha ,\gamma };d_n^{-1}{\overline{S}}_{N_n}⟹|T_{r,\alpha }{|}^{1/\gamma };d_n^{-1}{\underline{S}}_{N_n}⟹-{\left|T_{r,\alpha }\right|}^{1/\gamma };$$

$$d_n^{-1}|S_{N_n}|⟹|T_{r,\alpha }{|}^{1/\gamma };d_n^{-2}{\sum }_{N_n}^2⟹Y_{r,\alpha ,\gamma }^{*}.$$

Actually, Proposition 7 concerns not only the convergence of the distributions of maximum, minimum random sums or the absolute values of random sums to the generalized Burr distribution, but it also states that the condition $d_n^{-2}{\sum }_{N_n}^2⟹Y_{r,\alpha ,\gamma }^{*}$ is necessary and sufficient for the convergence of the distributions of random sums to the symmetric two-sided generalized Burr distribution introduced in Definition 1 (also see the equivalent representation (39)).

By virtue of Corollary 10, limit theorems for random sums with the symmetric two-sided Snedecor–Fisher distribution, as well as limit theorems for maximum, minimum random sums or the absolute values of random sums in which the limit laws is the Snedecor–Fisher distribution itself, can be obtained in exactly the same way as Proposition 7 was proven.

5.2. Generalized Burr Distribution as a Limit Law for Extreme Order Statistics in Samples with Random Sizes

In accordance with Proposition 2, the generalized Burr distributions with $0<\alpha \ge 2$ can be represented in the form of scale mixtures of the Weibull distribution, whereas these distributions with $1<\alpha \ge 2$ are representable as scale mixtures of the Fréchet distribution. In other words, relation (23) can be rewritten in the following form: for any $x\ge 0$

$$\mathsf{P}\left(\right|T_{r,\alpha }{|}^{1/\gamma }<x)={\int }_0^{\infty }(1-e^{-z{x}^{\gamma }})d\mathsf{P}\left({(U_{1,\alpha }\circ G_{r,r}^{1/\alpha })}^{1/\gamma }<z\right),$$

(50)

Relation (24) means that

$$\mathsf{P}\left(\right|T_{r,\alpha }{|}^{1/\gamma }<x)={\int }_0^{\infty }(1-e^{-z{x}^{\alpha \gamma }})d\mathsf{P}\left({(Z_{1/\alpha ,1}\circ G_{r,r})}^{1/\left(\alpha \gamma \right)}<z\right),$$

(51)

whereas relation (25) can be expressed as

$$\mathsf{P}\left(\right|T_{r,\alpha }{|}^{1/\gamma }<x)={\int }_0^{\infty }{e}^{-z{x}^{-\alpha \gamma }}d\mathsf{P}\left({(Z_{r,1}\circ G_{1/\alpha ,r})}^{-1/\left(\alpha \gamma \right)}<z\right).$$

(52)

As is well known, all the parent distributions in these mixtures can be limiting for extreme order statistics.

From (50)–(52) it follows that the generalized Burr distribution with $0<\alpha \le 2$ can be the limit distribution for extreme order statistics constructed from samples with random sizes. This fact can be illustrated by a special limit setting dealing with the max-compound and min-compound doubly stochastic Poisson processes.

Recall that a doubly stochastic Poisson process is a stochastic point process of the form $N\left(t\right)\stackrel{\mathrm{def}}{=}\mathsf{\Pi }\left(L\left(t\right)\right)$, where $\mathsf{\Pi }\left(t\right)$, $t\ge 0$ is a Poisson process with unit intensity and the stochastic process $L\left(t\right)$, $t\ge 0$ is independent of $\mathsf{\Pi }\left(t\right)$ and possesses the following properties: $L\left(0\right)=0$, $\mathsf{P}\left(L\right(t)<\infty )=1$ for any $t>0$, the sample paths of $L\left(t\right)$ are right-continuous and do not decrease. For more details concerning Cox and more general subordinated processes see, e.g., [34,35,36].

Let $T_1,T_2,\dots$ be the jump points of the process $N\left(t\right)$. Let ${\left\{(T_i,X_i)\right\}}_{i\ge 1}$ be a marked Cox point process, where the marks $X_1,X_2,\dots$ are independent identically distributed random variables independent of the process $N\left(t\right)$. In most studies dealing with the point process ${\left\{(T_i,X_i)\right\}}_{i\ge 1}$, the focus is made on traditional compound Cox process $S\left(t\right)$ that are defined as the sum of all marks $X_i$ of the points $T_i$ of the marked Cox point process which do not exceed the time t. In $S\left(t\right)$, the compounding operation is summation. At the same time, in many applied problems, it is very important to consider other functions of the marked Cox point process ${\left\{(T_i,X_i)\right\}}_{i\ge 1}$, for example, the so-called max-compound Cox process or min-compound Cox process that differ from $S\left(t\right)$ by the fact that the compounding operation of summation is replaced by the operation of taking the maximum or minimum of the marking random variables, respectively. For more detail concerning the analytic and asymptotic properties of max-compound and min-compound Cox processes, see [37,38,39].

Let $N\left(t\right)$ be a Cox process. The process $M\left(t\right)$ defined as

$$M\left(t\right)=\left\{\begin{array}{cc}-\infty ,\hfill & \mathrm{if}N\left(t\right)=0,\hfill \\ {\displaystyle \underset{1\le k\le N\left(t\right)}{\max }{X}_k,}\hfill & \mathrm{if}N\left(t\right)\ge 1,\hfill \end{array}\right.$$

$t\ge 0$, is called a max-compound Cox process.

The process $m\left(t\right)$ defined as

$$m\left(t\right)=\left\{\begin{array}{cc}+\infty ,\hfill & \mathrm{if}N\left(t\right)=0,\hfill \\ {\displaystyle \underset{1\le k\le N\left(t\right)}{\min }{X}_k,}\hfill & \mathrm{if}N\left(t\right)\ge 1,\hfill \end{array}\right.$$

$t\ge 0$, is called a min-compound Cox process.

By $F\left(x\right)$, we will denote the common distribution function of the random variables $X_j$. We will also use the conventional notation

$$\mathrm{lext}\left(F\right)=\inf \{x:F(x)>0\},\mathrm{rext}\left(F\right)=\sup \{x:F(x)<1\}.$$

Also, denote $P_F\left(x\right)\equiv F\left(\mathrm{lext}\left(F\right)-x^{-1}\right)$

Lemma 13.

Assume that there exist a positive infinitely increasing function $d\left(t\right)$ and a positive random variable L such that

$$\frac{L\left(t\right)}{d\left(t\right)}⟹L$$

(53)

as $t\to \infty$. Let also $\mathrm{lext}\left(F\right)>-\infty$ and let there exist a number $\delta >0$ such that for any $x>0$

$$\underset{y\to \infty }{\lim }\frac{P_F\left(yx\right)}{P_F\left(y\right)}=x^{-\delta }.$$

(54)

Then, there exist functions $a\left(t\right)$ and $b\left(t\right)$ such that

$$\mathsf{P}\left(\frac{m\left(t\right)-a\left(t\right)}{b\left(t\right)}<x\right)⟹H\left(x\right)$$

as $t\to \infty$, where

$$H\left(x\right)=\left\{\begin{array}{cc}{\displaystyle {\int }_0^{\infty }(1-e^{-z{x}^{\delta }})d\mathsf{P}(L<z),}\hfill & x\ge 0,\hfill \\ 0,\hfill & x<0.\hfill \end{array}\right.$$

Furthermore, the functions $a\left(t\right)$ and $b\left(t\right)$ can be defined as

$$a\left(t\right)=\mathrm{lext}\left(F\right),b\left(t\right)=\sup \left\{x:F\left(x\right)\le \frac{1}{d\left(t\right)}\right\}-\mathrm{lext}\left(F\right).$$

(55)

Proof. 

This lemma can be proven in exactly the same way as Theorem 2 in [37] dealing with max-compound Cox processes, with the account of the obvious relation

$$\min \{X_1,\dots ,X_{N\left(t\right)}\}=-\max \{-X_1,\dots ,-X_{N\left(t\right)}\}.$$

□

Proposition 8.

Let $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$ and $\gamma >0$. Assume that there exists a positive infinitely increasing function $d\left(t\right)$ such that condition (53) holds with

$$L\stackrel{d}{=}{(U_{1,\alpha }\circ G_{r,r}^{1/\alpha })}^{1/\gamma }.$$

Also, assume that $\mathrm{lext}\left(F\right)>-\infty$ and condition (54) holds with $\delta =\gamma$. Then, there exist functions $a\left(t\right)$ and $b\left(t\right)$ such that

$$\frac{m\left(t\right)-a\left(t\right)}{b\left(t\right)}⟹{\left|T_{r,\alpha }\right|}^{1/\gamma }$$

(56)

as $t\to \infty$. Furthermore, the functions $a\left(t\right)$ and $b\left(t\right)$ can be defined by (55).

Proof. 

This statement directly follows from Proposition 2, 1 and Lemma 13 with the account of (50). □

Proposition 9.

Let $1<\alpha \le 2$, $r>\frac{1}{\alpha }$ and $\gamma >0$. Assume that there exists a positive infinitely increasing function $d\left(t\right)$ such that condition (53) holds with

$$L\stackrel{d}{=}{(Z_{1/\alpha ,1}\circ G_{r,r})}^{1/\left(\alpha \gamma \right)}.$$

Also, assume that $\mathrm{lext}\left(F\right)>-\infty$ and condition (54) holds with $\delta =\alpha \gamma$. Then, there exist functions $a\left(t\right)$ and $b\left(t\right)$ such that

$$\frac{m\left(t\right)-a\left(t\right)}{b\left(t\right)}⟹{\left|T_{r,\alpha }\right|}^{1/\gamma }$$

(57)

as $t\to \infty$. Furthermore, the functions $a\left(t\right)$ and $b\left(t\right)$ can be defined by (55).

Proof. 

This statement directly follows from Proposition 2, II and Lemma 13 with the account of (51). □

Lemma 14.

Assume that there exist a positive infinitely increasing function $d\left(t\right)$ and a non-negative random variable L such that condition (53) holds. Also, assume that $\mathrm{rext}\left(F\right)=\infty$ and there exists a positive number δ such that

$$\underset{y\to \infty }{\lim }\frac{1-F\left(yx\right)}{1-F\left(y\right)}=x^{-\delta }$$

(58)

for any $x>0$. Then, there exist a positive function $b\left(t\right)$ and a distribution function $H_1\left(x\right)$ such that

$$\mathsf{P}\left(\frac{M\left(t\right)}{b\left(t\right)}<x\right)⟹H_1\left(x\right)$$

as $t\to \infty$. Furthermore,

$$H_1\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x<0,\hfill \\ {\displaystyle {\int }_0^{\infty }{e}^{-z{x}^{-\delta }}d\mathsf{P}(L<z)},\hfill & x\ge 0,\hfill \end{array}\right.$$

and the function $b\left(t\right)$ can be defined as

$$b\left(t\right)=\inf \left\{x:1-F\left(x\right)\le \frac{1}{d\left(t\right)}\right\}.$$

(59)

Proposition 10.

Let $1<\alpha \le 2$, $\frac{1}{\alpha }<r<1$ and $\gamma >0$. Assume that there exists a positive infinitely increasing function $d\left(t\right)$ such that condition (53) holds with

$$L\stackrel{d}{=}{(Z_{r,1}\circ G_{1/\alpha ,r})}^{-1/\left(\alpha \gamma \right)}.$$

Also, assume that $\mathrm{rext}\left(F\right)=\infty$ and condition (58) holds with $\delta =\alpha \gamma$. Then, there exists a positive function $b\left(t\right)$ such that

$$\frac{M\left(t\right)}{b\left(t\right)}⟹{\left|T_{r,\alpha }\right|}^{1/\gamma }$$

(60)

as $t\to \infty$. Furthermore, the function $b\left(t\right)$ can be defined by (59).

Proof. 

This statement directly follows from Proposition 2, III and Lemma 14 with the account of (52). □

As an example of processes satisfying the conditions described in Propositions 8–10, consider the following construction. Let $L\left(t\right)\equiv Yt$ and $d\left(t\right)\equiv t$, $t\ge 0$, where Y is a positive random variable. Then, the validity of the corresponding condition for the convergence of $L\left(t\right)/d\left(t\right)$ can be provided by an appropriate choice of Y. Moreover, t may not have the meaning of the physical time. For example, it may be a location parameter of $L\left(t\right)$ (mean or median). Actually, the statements presented in this section deal with the case of large mean intensity of the Cox process.

By virtue of Corollary 11, limit theorems for extreme order statistics in samples with random size with the Snedecor–Fisher limit distribution can be obtained in exactly the same way as Propositions 9 and 10 were proven.

5.3. M-Asymmetric Two-Sided Generalized Burr and M-Asymmetric Generalized Student Distributions as Limit Laws for Random Sums

Among many possible mixture representations for the generalized Student distribution (see [1]) normal mixture model seems most promising for the construction of asymmetric generalizations since it presents the opportunity to formulate a rather simple version of the central limit theorem (more exactly, a special transfer theorem) for random sums of independent identically distributed random variables in which the M-asymmetric generalized Student and M-asymmetric two-sided generalized Burr distributions act as the limit laws. Moreover, the conditions imposed on the distribution of the summands are rather loose. For example, although the limit distributions appear to be heavy-tailed, the summed random variables may have finite variance and may even be bounded. The corresponding statement is based on the transfer theorem for random sums in the ‘if and only if’ form (see, e.g., [40]) and the identifiability of normal variance–mean mixtures [41].

Let ${\left\{X_{n,j}\right\}}_{j\ge 1},n=1,2,\dots$ be a double array of row-wise (that is, for each fixed n) identically distributed and independent random variables. Let ${\left\{N_n\right\}}_{n\ge 1}$ be a sequence of non-negative integer-valued random variables such that for each $n\ge 1$, the random variables $N_n,X_{n,1},X_{n,2},\dots$ are independent. Let

$$S_{n,k}=X_{n,1}+\dots +X_{n,k}.$$

For definiteness, we assume ${\sum }_{j=1}^0=0$. The symbol ⟹ will denote convergence in distribution. In the statement below, the convergence is meant as $n\to \infty$.

Lemma 15

([41]). Assume that there exist a sequence of natural numbers ${\left\{k_n\right\}}_{n\ge 1}$ and numbers $a\in \mathbb{R}$ and $b>0$ such that

$$\mathsf{P}\left(S_{n,k_n}<x\right)⟹\mathsf{\Phi }\left(\frac{x-a}{b}\right).$$

(61)

Assume that $N_n\to \infty$ in probability (that is, $\mathsf{P}(N_n\ge K)\to 0$ for any $K\in (0,\infty )$). The distributions of random sums $S_{N_n}$ converge to some distribution function $F\left(x\right)$:

$$\mathsf{P}\left(S_{n,N_n}<x\right)⟹F\left(x\right),$$

if and only if there exists a distribution function $H\left(x\right)$ such that $H\left(0\right)=0$,

$$F\left(x\right)={\int }_0^{\infty }\mathsf{\Phi }\left(\frac{x-az}{b\sqrt{z}}\right)dH\left(z\right),$$

(62)

and

$$\mathsf{P}(N_n<x{k}_n)⟹H\left(x\right).$$

It is easily seen that expression (41) defining the M-asymmetric generalized Student distribution coincides with (62), where $H\left(z\right)=\mathsf{P}(D_{r,\alpha }<z)$. Hence, Lemma 15 directly yields the following result.

Proposition 11.

Let $\alpha \in (0,2]$, $r>0$, $a\in \mathbb{R}$, $\sigma >0$. Assume that there exist a sequence of natural numbers ${\left\{k_n\right\}}_{n\ge 1}$ and numbers $a\in \mathbb{R}$ and $b>0$ such that condition (61) holds. Assume that $N_n\to \infty$ in probability. Then, the distributions of random sums $S_{N_n}$ converge to the M-asymmetric generalized Student distribution function ${\tilde{F}}_{r,\alpha ,a,\sigma }\left(x\right)$(see (41))

$$\mathsf{P}\left(S_{n,N_n}<x\right)⟹{\tilde{F}}_{r,\alpha ,a,\sigma }\left(x\right),$$

if and only if

$$\mathsf{P}(N_n<x{k}_n)⟹\mathsf{P}(D_{r,\alpha }<x).$$

Similarly, expression (43) defining the M-asymmetric two-sided generalized Burr distribution coincides with (62), where $H\left(z\right)=\mathsf{P}(Y_{r,\alpha ,\gamma }^{*}<z)$. Hence, Lemma 15 directly yields the following result.

Proposition 12.

Let $\alpha \in (0,1]$, $r>\frac{1}{\alpha }$, $\gamma \in (0,1]$, $a\in \mathbb{R}$, $\sigma >0$. Assume that there exist a sequence of natural numbers ${\left\{k_n\right\}}_{n\ge 1}$ and numbers $a\in \mathbb{R}$ and $b>0$ such that condition (61) holds. Assume that $N_n\to \infty$ in probability. The distributions of random sums $S_{N_n}$ converge to the M-asymmetric two-sided generalized Burr distribution function ${\tilde{F}}_{r,\alpha ,a,\sigma }\left(x\right)$(see (43))

$$\mathsf{P}\left(S_{n,N_n}<x\right)⟹{\tilde{A}}_{r,\alpha ,\gamma ,a,\sigma }\left(x\right),$$

if and only if

$$\mathsf{P}(N_n<x{k}_n)⟹\mathsf{P}(Y_{r,\alpha ,\gamma }^{*}<x).$$

Propositions 11 and 12 may serve as the theoretic explanation of the possible utility of the M-asymmetric generalized Student and the M-asymmetric two-sided generalized Burr distribution as convenient heavy-tailed models for statistical regularities observed in some real phenomena in which the additive structure of the observed data can be assumed. In [1], we have already discussed this question. Although the distributions of $D_{r,\alpha }$ or $Y_{r,\alpha ,\gamma }^{*}$ may seem to be rather curious, there are no serious objections against the possibility of application of these distributions for modelling poorly predictable regularities of, say, information flows in financial markets. For more details see [1].

In exactly the same way, by virtue of (26) and Corollary 10, we can obtain the limit theorem for random sums in which the limit law is the M-asymmetric two-sided Snedecor–Fisher distribution.