 Research
 Open Access
Analytical properties of generalized Gaussian distributions
 Alex Dytso^{1}Email authorView ORCID ID profile,
 Ronit Bustin^{2},
 H. Vincent Poor^{1} and
 Shlomo Shamai^{2}
https://doi.org/10.1186/s4048801800885
© The Author(s) 2018
 Received: 20 March 2018
 Accepted: 4 November 2018
 Published: 4 December 2018
Abstract
The family of Generalized Gaussian (GG) distributions has received considerable attention from the engineering community, due to the flexible parametric form of its probability density function, in modeling many physical phenomena. However, very little is known about the analytical properties of this family of distributions, and the aim of this work is to fill this gap.
Roughly, this work consists of four parts. The first part of the paper analyzes properties of moments, absolute moments, the Mellin transform, and the cumulative distribution function. For example, it is shown that the family of GG distributions has a natural order with respect to secondorder stochastic dominance.
The second part of the paper studies product decompositions of GG random variables. In particular, it is shown that a GG random variable can be decomposed into a product of a GG random variable (of a different order) and an independent positive random variable. The properties of this decomposition are carefully examined.
The third part of the paper examines properties of the characteristic function of the GG distribution. For example, the distribution of the zeros of the characteristic function is analyzed. Moreover, asymptotically tight bounds on the characteristic function are derived that give an exact tail behavior of the characteristic function. Finally, a complete characterization of conditions under which GG random variables are infinitely divisible and selfdecomposable is given.
The fourth part of the paper concludes this work by summarizing a number of important open questions.
Keywords
 Generalized Gaussian distribution
 Infinite divisibility
 Mellin transform
 Characteristic function
 Selfdecomposition
Introduction
respectively. Another commonly used name for this type of distribution, especially in economics, is the Generalized Error distribution. The flexible parametric form of the pdf of the GG distribution allows for tails that are either heavier than Gaussian (p<2) or lighter than Gaussian (p>2) which makes it an excellent choice for many modeling scenarios. The origin of the GG family can be traced to the seminal work of Subbotin (1923) and Lévy (1925). In fact, Subbotin (1923) has shown that the same axioms used by Gauss (1809) to derive the normal distribution, are also satisfied by the GG distribution. Wellknown examples of this distribution include: the Laplace distribution for p=1; the Gaussian distribution for p=2; and the uniform distribution on [μ−α,μ+α] for p=∞.
1.1 Past work
The GG distribution has found use in image processing applications where many statistical features of an image are naturally modeled by distributions that are heaviertailed than Gaussian.
For example, Gabor coefficients are convolution kernels whose frequency and orientation representations are similar to those of the human visual system. Gabor coefficients have found a wide range of applications in texture retrieval and facerecognition problems. However, a considerable drawback of using Gabor coefficients is the memory requirements needed to store a Gabor representation of an image. In GonzalezJimenez et al. (2007) GG distributions with the parameter p<2 have been shown to accurately approximate the empirical distribution of Gabor coefficients in terms of the KullbackLiebler (KL) divergence and the χ^{2} distance. Moreover, the authors in (GonzalezJimenez et al. 2007) demonstrated that data compression algorithms based on the GG statistical model considerably reduce the memory required to store Gabor coefficients.
In a classical image retrieval problem, a system searches for K images similar to a query image from a digital library containing a total of N images (usually K≪N). In (Do and Vetterli 2002) by modeling wavelet coefficients with a GG distribution and using the KL divergence as a similarity measure, the authors were able to improve retrieval rates by 65% to 70%, compared with traditional approaches.
Other applications of the GG distribution in image processing applications include modeling: textured images, see Mallat (1989); Moulin and Liu (1999) and de Wouwer et al. (1999); pixels forming fineresolution synthetic aperture radar (SAR) images (Bernard et al. 2006); and the distribution of values in subband decompositions of video signals Westerink et al. (1991) and Sharifi and LeonGarcia (1995).
In communication theory, the GG distribution finds many modeling applications in impulsive noise channels which occur when the noise pdf has a longer tail than the Gaussian pdf. For example, in Beaulieu and Young (2009) it is shown that in ultrawideband (UWB) systems with timehopping (TH) the interference should be modeled with probability distributions that are more impulsive than the Gaussian. Moreover, it has been shown that for the moderate and high signaltonoise ratio (SNR) the interference in the THUWB is well modeled by the GG distribution with a parameter p≤1. In Algazi and Lerner (1964) and Miller and Thomas (1972) certain atmospheric noises were shown to be impulsive and GG distributions with parameter values of 0.1<p<0.6 were shown to provide good approximations to their distributions.
GG distributions can also model noise distributions that appear in nonstandard wireless media. In Nielsen and B.Thomas (1987) the authors showed that Arctic underice noise is well modeled by members of the GG family. In Banerjee and Agrawal (2013) the GG family has been recognized as a model for the underwater acoustic channel where values of p=2.2 and p=1.6 have been found to model the ship transit noise and the sea surface agitation noise, respectively.
The problem of designing optimal detectors for signals in the presence of GG noise has been considered in Miller and Thomas (1972); Poor and Thomas (1978) and Viswanathan and Ansari (1989). In Soury et al. (2012) the authors studied the average bit error probability of binary coherent signaling over flat fading channels subject to additive GG noise. Interestingly, the authors of Soury et al. (2012) give an exact expression for the average probability of error in terms of Fox’s H functions.
In power systems, the GG distribution has been used to model hourly peak load model demand in power grids (Mohamed et al. 2008).
In Varanasi and Aazhang (1989) the authors studied a problem of estimating parameters of the GG distribution (order p, mean μ, and variance \(\sigma ^{2}=\mathbb {E}\left [(X_{p}\mu)^{2}\right ]\)) from n independent realizations of a GG random variable. The authors of (Varanasi and Aazhang 1989) considered three estimation methods, namely, the method of moments, maximum likelihood, and moment/Newtonstep estimators, and compared performance of each for different values of p. For example, in the vicinity of p=2, the moment method was shown to perform best. In (Richter 2007) the authors established connections between chisquare and Student’s tdistribution. Moreover, in Richter (2016), using the notions of generalized chisquare and Fisher statistics introduced in Richter (2007), the authors studied a problem of inferring one or two scaling parameters of the GG distribution and derived both the confidence interval and significance test.
The Shannon capacity of channels with GG noise has been considered in Fahs and AbouFaycal (2018) and Dytso et al. (2017b). In Fahs and AbouFaycal (2018) the authors gave general results on the structure of the optimal input distribution in channels with GG noise under a large family of channel input cost constraints. In Dytso et al. (2017b) the authors investigated the capacity of channels with GG noise under L_{p} moment constraints and proposed several upper and lower bounds that are asymptotically tight.
As the pdf of GG distributions has a very simple form, many quantities such as moments, entropy, and Rényi entropy can be easily computed (Do and Vetterli 2002; Nadarajah 2005). Also, from the information theoretic perspective the GG distribution is interesting because it maximizes the entropy under a pth absolute moment constraint (Cover and Thomas 2006; Lutwak et al. 2007). The maximum entropy property can serve as an important intermediate step in a number of proofs. For example, in (Dytso et al. 2018) it has been used to generalize the OzarowWyner bound (Ozarow and Wyner 1990) on the mutual information of discrete inputs over arbitrary channels. In Nielsen and Nock (2017) the maximum entropy principle has been used to improve bounds on the entropy of Gaussian mixtures.
While the number of applications of the GG distribution is large, many of its properties have been drawn from numerical studies, and few analytical properties of the GG family are known beyond the cases p=1,2 and p=∞. For instance, very little is known about the characteristic function of the GG distribution and only expressions in terms of hypergeometric functions are known. For example, the characteristic function of the GG distribution was given in terms of FoxWrite functions in Pogány and Nadarajah (2010) for all p>1 and later generalized in terms of FoxH functions in Soury and Alouini (2015) for all p>0. The work of Soury and Alouini (2015), also characterized the pdf of the sum of two independent GG random variables in terms of FoxH functions. Specific nonlinear transformations of sums of independent GG distributions and the moment generating function of the GG distribution have been studied in Vasudevay and Kumari (2013).
There is also a large body of work on multivariate GG distributions. For example, to the best of our knowledge, the first multivariate generalization was introduced in (De Simoni 1968) where the exponent was taken to be \( \left (\left (\textbf {x} \boldsymbol {\mu }\right)^{T} \textbf {K}^{1} (\textbf {x}\boldsymbol {\mu }) \right)^{\frac {p}{2}}\) where x and μ are vectors and K is a matrix. In Goodman and Kotz (1973) the authors introduced yet another multivariate generalization of the GG distribution in (1): X is said to be multivariate GG if and only if it can be written as X=KZ+μ where the components of Z are independently and identically distributed according to the univariate GG distribution in (1). An example of multivariate distributions with GG marginals and examples of multivariate GG distributions defined with respect to other norms the interested reader is referred to Richter (2014); ArellanoValle and Richter (2012) and Gupta and Nagar (2018) and the references therein.
1.2 Paper outline and contributions
 1In “Moments and the Mellin transform” section, we study properties of the moments of the GG distribution including the following:

In Proposition 1 we derive an expression for the Mellin transform of the GG distribution; and

In Proposition 2 we show necessary and sufficient conditions under which moments of the GG distribution uniquely determine the distribution.

 2In “Properties of the distribution” section, we study properties of the distribution including the following:

In “Stochastic ordering” section, Proposition 3 shows that the family of GG distributions is an ordered set where the order is taken in terms of secondorder stochastic dominance; and

In “Relation to completely monotone functions and positive definiteness” section, Theorem 1 connects the pdf of GG distributions to positive definite functions. In particular, we show that for p≤2 the pdf of the GG distribution is a positive definite function and for p>2 the pdf is not a positive definite function. Moreover, it is shown that for p≤2 the pdf of the GG distribution can be expressed as an integral of a Gaussian pdf with respect to a nonnegative finite Borel measure.

 3In “On product decomposition of GG random variables” section, Proposition 5 shows that the GG random variable X_{p} can be decomposed into a product of two independent random variables X_{p}=V·X_{r} where X_{r} is a GG random variable. We carefully study properties of this decomposition including the following:

In “On the PDF of V_{p,q}” section, Proposition 6 gives power series and integral representations of the pdf of V; and

In “On the determinacy of the distribution of V_{G,q}” section, Proposition 8 shows under which conditions the distribution of V is completely determined by its moments. Interestingly, the range for values of p for which X_{p} and V are determinant is not the same. This gives an interesting example that the product of two determinate random variables is not necessarily determinate.

 4In “Characteristic function” section, we study properties of the characteristic function of the GG distribution including the following:

In “Connection to stable distributions” section, Proposition 9 discusses connections between a class of GG distributions and a class of symmetric stable distributions;

In “Analyticity of the characteristic function” section, Proposition 10 shows under what conditions the characteristic function of the GG distribution is a real analytic function;

In “On the distribution of zeros of the characteristic function” section, Theorem 3 studies the distribution of zeros of the characteristic function of the GG distribution. In particular, it is shown that for p≤2 the characteristic function of the GG distribution has no zeros and is always positive, and for p>2 the characteristic function has at least one positivetonegative zero crossing; and

In “Asymptotic behavior of ϕ_{p}(t)” section, Proposition 11 gives the tail behavior of the characteristic function of the GG distribution and its derivatives. The consequences of this result are discussed.

 5In “Additive decomposition of a GG random variable” section, we study additive decompositions of the GG random variables including the following:

In “Infinite divisibility of the characteristic function” section, Theorem 5 completely characterizes for which values of p the GG random variable is infinitely divisible. In addition, Proposition 14 studies properties of the canonical LévyKhinchine representation of infinitely divisible distributions; and

In “Selfdecomposability of the characteristic function” section, Theorem 6 characterizes conditions under which a GG distribution of order p can be additively transformed into another GG distribution of order q. In the case of p=q this corresponds to answering if a GG distribution is selfdecomposable.

The paper is concluded in “Discussion and conclusion” section by reflecting on future directions.
1.3 Other parametrization of the PDF
In the above parametrization the pth moment, when μ=0, is normalized such that it equals to σ^{p}.
The choice of the parametrization is usually dictated by the application that one has in mind. In this work, we choose to work with the parametrization in (1) which we found to be convenient for studying the Mellin transform and the characteristic function of the GG distribution.
Moments and the Mellin transform
In this section, we study properties of the moments, absolute moments and Mellin transform of the GG distribution. We also show conditions under which the moments of X_{p} uniquely characterize its distribution. While the majority of the results in this section are not new or are easy to derive, we choose to include them for completeness as most of the development in other section will heavily depend on properties of moments.
2.1 Moments, absolute moments, and the Mellin transform
Definition 1
Proposition 1
Proof
where in the last step we used the value of c_{p} in (1). Moreover, the above integral is finite if Re(s)>0 and p>0. The proof of (11) follows by choosing s=k+1 in (10). This concludes the proof. □
Note that the pth absolute moment of X_{p} is given by \(\mathbb {E}\left [\left X_{p}\right ^{p}\right ]= \frac {2\alpha ^{p}}{p}.\)
The expression in (11) can also be extended to multivariate GG distributions defined through ℓ_{p} norms; see for example Lutwak et al. (2007) and ArellanoValle and Richter (2012).
The following corollary, which relates kth moments of two GG distributions of a different order, is useful in many proofs.
Corollary 1
Proof
See Appendix A. □
2.2 Moment problem
The classical moment problem asks whether a distribution can be uniquely determined by its moments. For random variables defined on \(\mathbb {R}\), this problem goes under the name of the Hamburger moment problem and for random variables on \(\mathbb {R}^{+}\) under the name of the Stieltjes moment problem (Stoyanov 2000). If the answer is affirmative, we say that the moment problem is determinate. Otherwise, we say that the moment problem is indeterminate and there exists another distribution that shares the same moments.
Proposition 2
The GG distribution is determinate for p∈[1,∞) and indeterminate for p∈(0,1).
Proof
To show that the distribution is determinate it is enough to show that the characteristic function has a power series expansion with a positive radius of convergence. For the GG distribution with p∈[1,∞), this will be done in Proposition 10. □
The interested reader is referred to [Lin and Huang (1997), Theorem 2] and [HoffmanJørgensen (2017), p. 301] where the conditions for the moment determinacy are provided for a Double Generalized Gamma distribution of which a GG distribution is special case.
Remark 1
Remark 2
In (Varanasi and Aazhang 1989) the authors studied the problem of estimating the parameter p from n independent realizations of a GG random variable. As one of the proposed methods, the authors used empirical moments to estimate the parameter p. Moreover, in Varanasi and Aazhang (1989) it has been observed that the method of moments performs poorly for p∈(0,1). In view of Proposition 2, the observation about the method of moments made in Varanasi and Aazhang (1989) can be attributed to the fact that the GG distribution is indeterminate for p∈(0,1).
Properties of the distribution
3.1 Stochastic ordering
Corollary 1 suggests that there might be some ordering between members of the GG family. To make this point more explicit we need the following definition.
Definition 2
Proposition 3
Let \(X_{p}\sim \mathcal {N}_{p}(0,1)\) and \(X_{q}\sim \mathcal {N}_{q}(0,1)\). Then, for p≤q, X_{q} dominates X_{p} in the sense of the secondorder stochastic dominance.
Proof
See Appendix B. □
From Proposition 3 we have the following inequality for the expected value of functions of GG distributions.
Proposition 4
Proof
The inequality in (19) is equivalent to the secondorder stochastic dominance. For more details, the interested reader is referred toLevy (1992). □
In particular, the inequality in (21) shows that the Laplace transform of \(f_{X_{p}}\) (which exists if 1<p,q) is larger than the Laplace transform of \(f_{X_{q}}\).
3.2 Relation to completely monotone functions and positive definiteness
We begin by introducing the notion of completely monotone and Bernstein functions.
Definition 3
A function f:[0,∞)→[0,∞) is said to be a Bernstein function if the derivative of f is a completely monotone function.
Applying the wellknown result fromSchilling et al. (2012), that the composition of a completely monotone function and a Bernstein function is completely monotone, on the function e^{−x} (completely monotone) and the function \(\frac {x^{p}}{2}\) (Bernstein for p∈(0,1]) we obtain the following.
Corollary 2
For p∈(0,1] the function \( \mathrm {e}^{\frac {x^{p}}{ 2 }}\) is completely monotone.
For p>1 the function \(\mathrm {e}^{x^{p}}\) is not completely monotone.
As will be observed throughout this paper, the GG distribution exhibits different properties depending on whether p≤2 or p>2. At the heart of this behavior is the concept of positivedefinite functions.
Definition 4
is positive semidefinite.
The next result relates the pdf of the GG distribution to the class of positive definite functions.
Theorem 1

not positive definite for p∈(2,∞); and

positive definite for p∈(0,2]. Moreover, there exists a finite nonnegative Borel measure μ_{p} on \(\mathbb {R}^{+}\) such that for x>0$$ \mathrm{e}^{\frac{x^{p}}{2}}= \int_{0}^{\infty} e^{\frac{t}{2}x^{2}} d\mu_{p}(t). $$(24)
Proof
See Appendix C. □
The expression in (24) will form a basis for much of the analysis in the regime p∈(0,2] and will play an important role in examining properties of the characteristic function of the GG distribution. The following corollary of Theorem 1 will also be useful.
Corollary 3
Proof
The proof follows by substituting x in (24) with \(x^{\frac {q}{p}}\). □
On product decomposition of GG random variables
As a consequence of Theorem 1 we have the following decompositional representation of the GG random variable.
Proposition 5

V_{p,q} is an unbounded random variable for p<2 and V_{p,q}=1 for p=2; and

for p<2, V_{p,q} is a continuous random variable with pdf given by$$ f_{V_{p,q}}(v)= \frac{1}{2\pi} \frac{\Gamma \left(\frac{p}{2q} \right)}{\Gamma \left(\frac{1}{q} \right)} \int_{\mathbb{R}} v^{it1} \frac{2^{\frac{it}{q}} \Gamma \left(\frac{it +1}{q}\right)}{2^{\frac{itp}{2q}} \Gamma \left(\frac{p(it+1)}{2q}\right)} dt, \; v>0. $$(26b)
Proof
See Appendix D. □
Proposition 5 can be used to show that the GG random distribution is a Gaussian mixture which is formally defined next.
Definition 5
A random variable X is called a (centered) Gaussian mixture if there exists a positive random variable V and a standard Gaussian random variable Z, independent of V, such that X=dVZ.
As a consequence of Proposition 5 we have the following result.
Corollary 4
where V_{q,q} is independent of X_{2} and its pdf is defined in (26b).
Proof
The proof follows by choosing p=q in (26a). □
respectively.
4.1 On the PDF of V _{p,q}
The expression for the pdf of V_{p,q} in (26b) can be difficult to analyze due to the complex nature of the integrand. The next result provides two new representations of the pdf of V_{p,q} that in many cases are easier to analyze than the expression in (26b).
Proposition 6
 1Power Series Representation$$ f_{V_{p,q}}(v)= \frac{ \Gamma \left(\frac{p}{2q} \right)}{ \Gamma \left(\frac{1}{q} \right)} \sum\limits_{k=1}^{\infty} a_{k} v^{kq}, \ v>0, $$(28)where$$ a_{k}= \frac{q}{\pi} \frac{(1)^{k+1} 2^{(kq+1) \left(\frac{p}{2q} \frac{1}{q} \right)} \Gamma\left(\frac{kq}{2} +1 \right) \sin \left(\frac{\pi kq}{2} \right) }{k! }. $$(29)
 2Integral Representation$$ f_{V_{p,q}}(v)=\frac{q 2^{\frac{p}{2q}\frac{1}{q}} \Gamma \left(\frac{p}{2q} \right)}{ \pi \Gamma \left(\frac{1}{q} \right)} \int_{0}^{\infty} \sin \left(a_{p} v^{q} x^{\frac{p}{2}} \right) \mathrm{e}^{b_{p} v^{q} x^{\frac{p}{2}}x} dx, $$(30)where$$ a_{p}=2^{\frac{p}{2}1} \sin \left(\frac{\pi p}{2} \right), b_{p}=2^{\frac{p}{2}1} \cos \left(\frac{\pi p}{2} \right). $$(31)
Proof
See Appendix E. □
Remark 3
Proposition 7
Proof
The proof follows by taking the limit as v→0 in (34). □
As we will demonstrate later, the behavior of the pdf of V_{G,q} around zero will be important in studying the asymptotic behavior of the characteristic function of X_{q}. This is reminiscent of the initial value theorem of the Laplace transform where the value of a function at zero can be used to estimate the asymptotic behavior of its Laplace transform. Indeed, as we will see, the characteristic function of X_{q} and the Laplace transform of \(V_{G,q}^{2}\) have a clear connection.
4.2 On the determinacy of the distribution of V _{G,q}
Similar to the investigation in “Moment problem” section of whether GG distributions are determinant (uniquely determined by their moments) or not, we now conduct a similar investigation of the distributions of V_{G,q}.
Proposition 8
The distribution of V_{G,q} is determinant for \(q\ge \frac {2}{5}\).
Proof
By using conditions for the convergence of pseries the sum in (37) diverges if \( \frac {1}{2} \left (\frac {1}{q}\frac {1}{2} \right) \ge 1\) or \(q \ge \frac {2}{5}\). Therefore, Carleman’s condition is satisfied if \(q \ge \frac {2}{5}\), and thus V_{G,q} has a determinant distribution for \(q \ge \frac {2}{5}\). This concludes the proof. □
Remark 4
According to Proposition 2 and 8, for the range of values \(q \in \left [\frac {2}{5}, 1\right ]\) the random variable X_{q}=dV_{G,q}·X_{2} is a product of two random variables with determinant distributions while X_{q} itself has an indeterminate distribution on \(q \in \left [\frac {2}{5}, 1\right ]\) by Proposition 2. This observation generates an interesting example illustrating that the product of two independent random variables with determinant distributions can have an indeterminate distribution.
Characteristic function
The focus of this section is on the characteristic function of the GG distribution. The characteristic function of the GG distribution can be written in the following integral forms.
Theorem 2

For any p>0$$ \phi_{p}(t) = 2c_{p} \int_{0}^{\infty} \cos(t x) e^{\frac{x^{p}}{2}} dx, \, t \in \mathbb{R}. $$(38a)

For any p∈(0,2]$$ \phi_{p}(t) = \mathbb{E} \left[ \mathrm{e}^{\frac{t^{2} V_{G,p}^{2}}{2}} \right], \, t \in \mathbb{R}, $$(38b)
where the density of a variable V_{G,p} is defined in Proposition 5.
Proof
The proof of (38a) follows from the fact that \(e^{\frac {x^{p}}{2} }\) is an even function which implies that the Fourier transform is equivalent to the cosine transform.
The following result is immediate by Theorem 2.
Corollary 5
For p∈(0,2], ϕ_{p}(t) is a decreasing function for t>0.
5.1 Connection to stable distributions
A class of distributions that is closed under convolution of independent copies is called stable. A more precise definition is given next.
Definition 6
where ϕ_{X}(t) is a characteristic function of a random variable X.
Throughout this work we will use stable distribution, stable random variable, and stable characteristic function interchangeably.
where \(\mu \in \mathbb {R}\) is the shiftparameter, \(c \in \mathbb {R}^{+}\) is the scaling parameter, β∈[−1,1] is the skewness parameter, and α∈(0,2] is the order parameter. We refer the interested reader to (Zolotarev 1986) for a comprehensive treatment of the subject of stable distributions.
Observe that there is a duality between a class of symmetric stable distributions and a class of GG distributions with p∈(0,2]. Up to a normalizing constant, the pdf of a GG random variable is equal to the characteristic function of an αstable random variable. Equivalently, the pdf of an αstable random variable is equal, up to a normalizing constant, to the characteristic function of a GG random variable.
We exploit this duality to give, yet another, integral representation of the characteristic function of the GG distribution with parameter p∈(0,2].
Proposition 9

U_{p}(x) is a nonnegative function;

For p∈(0,1), U_{p}(x) is an increasing function with$${\lim}_{x \to 0^{+}} U_{p}(x)=0, \, {\lim}_{x \to 1^{}} U_{p}(x)=\infty; $$

For p∈(1,2], U_{p}(x) is a decreasing function with$${\lim}_{x \to 0^{+}} U_{p}(x)=\infty, \, {\lim}_{x \to 1^{}} U_{p}(x)=0; $$

For all p∈(0,2]∖{1}$${\lim}_{x \to 0^{+} }g_{p}(x)=0, \, {\lim}_{x \to 1^{}} g_{p}(x)=0; \text{ and} $$

The function g_{p} has a single maximum given by$$\max_{x \in [0,1]} g_{p}(x)= \frac{1}{ \mathrm{e} t^{\frac{p}{p1} }}. $$
Proof
The characterization in (43a) can be found in (Zolotarev 1986, Theorem 2.2.3). The proof of the properties of U_{p}(x) is presented in Appendix F. □
We suspect that most of the properties of ϕ_{p}(t) for p∈(0,2) that we derive in this paper can be found by using the integral expression in (43a). However, instead of taking this route we use the product decomposition in Proposition 5 to derive all the properties of ϕ_{p}(t). We believe that using a product decomposition is a more natural approach. Moreover, the positive random variables in Gaussian mixtures, V_{G,p} in our case, naturally appear in a number of applications (e.g., bounds on the entropy of sum of independent random variables (Eskenazis et al. 2016)) and are of independent interest.
5.2 Analyticity of the characteristic function
The above expression is especially useful since the moments of GG distributions are known for every k; see Proposition 1.
Proposition 10

\(t \in \mathbb {R}\) for p>1; and

\( t < \frac {1}{2} \) for p=1.
For p<1 the function ϕ_{p}(t) is not real analytic.
Proof
See Appendix G. □
The results of Proposition 10 also lead to the conclusion that for p>1 the moment generating function of X_{p}, \(M_{p}(t)=\mathbb {E}\left [e^{{tX}_{p}}\right ]\) exists for all \(t\in \mathbb {R}\).
5.3 On the distribution of zeros of the characteristic function
As seen from Fig. 2 the characteristic function of the GG distribution can have zeros. The next theorem gives a somewhat surprising result on the distribution of zeros of ϕ_{p}(t).
Theorem 3

for p>2, ϕ_{p}(t) has at least one positive to negative zero crossing. Moreover, the number of zeros is at most countable; and

for p∈(0,2], ϕ_{p}(t) is a positive function.
Proof
See Appendix H. □
Also, we conjecture that zeros of ϕ_{p}(t) have the following additional property.
Conjecture 1
For p∈(2,∞) zeros of ϕ_{p}(t) do not appear periodically.
It is important to point out that, for p=∞, the characteristic function is given by \(\phi _{\infty }(t)= \frac {\sin (t)}{t}=\text {sinc}(t)\), and zeros do appear periodically. However, for p<∞ we conjecture that zeros do not appear periodically.
5.4 Asymptotic behavior of ϕ _{p}(t)
Proposition 11
Proof
See Appendix I. □
Using Proposition 11, we can give an exact tail behavior for ϕ_{p}(t).
Proposition 12
Proof
The proof follows immediately from Proposition 11. □
Note that, for p∈(0,2], the function \(\phi _{p}(\sqrt {2t})\) can be thought of as a Laplace transform of the pdf of the random variable \(V_{G,p}^{2}\). This observation together with the asymptotic behavior of ϕ_{p}(t) leads to the following result.
Proposition 13
For \(n\in \mathbb {R}\), \(\mathbb {E}[V_{G,p}^{n}]\) is finite if and only if n+p>−1.
Proof
For n>−1 the proof is a consequence of the decomposition property in Propositions 5 and 1 where it is shown that \(\mathbb {E}[X_{p}^{n}]<\infty \) if n>−1 for all p>0. Therefore, we assume that n<−1.
Note that the integral in (49) is finite if and only if \( \phi _{p}\left (\sqrt {2t}\right) t^{k1}= O \left (t^{(1+\epsilon)}\right)\) for every ε>0. Moreover, by Proposition 12 we have that \(\phi _{p}\left (\sqrt {2t}\right) t^{k1}= O \left (\frac {t^{k1}}{t^{\frac {p+1}{2}}} \right)\), which implies that the integral in (49) is finite if and only if 2k−p<1. Setting 2k=−n concludes the proof. □
Additive decomposition of a GG random variable
In this section we are interested in determining whether a GG random variable \(X_{q}~\sim ~\mathcal {N}_{q}(0,\alpha ^{q})\) can be decomposed into a sum of two or more independent random variables.
6.1 Infinite divisibility of the characteristic function
Definition 7
Similarly to stable distributions, we use infinitely divisible distribution, infinitely divisible random variable, and infinitely divisible characteristic function interchangeably.
Next we summarize properties of infinitely divisible distributions needed for our purposes.
Theorem 4
 1
((Lukacs 1970, Theorem 5.3.1).) An infinitely divisible characteristic function has no real zeros;
 2
((van Harn and Steutel 2003, Theorem 10.1).) A symmetric distribution that has a completely monotone pdf on (0,∞) is infinitely divisible;
 3(LévyKhinchine canonical representation (Lukacs 1970, Theorem 5.5.1).) The function ϕ(t) is an infinitely divisible characteristic function if and only if it can be written as$$ \log \left(\phi(t) \right)= ita + \int_{\infty}^{\infty} \left(\mathrm{e}^{itx}1 \frac{itx}{1+x^{2}} \right) \frac{1+x^{2}}{x^{2}}d\theta(x), $$(51)
where a is real and where θ(x) is a nondecreasing and bounded function such that \({\lim }_{x \to \infty } \theta (x)=0\). The function dθ(x) is called the Lévy measure. The integrand is defined for x=0 by continuity to be equal to \(\frac {t^{2}}{2}\). The representation in (51) is unique; and
 4((van Harn and Steutel 2003, Corollary 9.9).) A nondegenerate infinitely divisible random variable X has a Gaussian distribution if and only if it satisfies$$ \limsup_{x \rightarrow \infty} \frac{ \log \mathbb{P}[ X \ge x] }{x \, \log (x)}=\infty. $$(52)
In general, the Lévy measure dθ is not a probability measure and hence the distribution function θ(x) is not bounded by one.
We use Theorem 4 to give a complete characterization of the infinite divisibility property of the GG distribution.
Theorem 5
A characteristic function ϕ_{p}(t) is infinitely divisible if and only if p∈ (0,1] ∪{2}.
Proof
For the regime p∈(0,1] in Corollary 2 it has been shown that the pdf is completely monotone on (0,∞). Therefore, by property 2) in Theorem 4 it follows that ϕ_{p}(t) is infinitely divisible for p∈(0,1].
where the equalities follow from: a) the expression for the CDF in (16); and b) using the limit \({\lim }_{x \to \infty } \frac {\Gamma (s,x)}{x^{s1} \mathrm {e}^{x} }=1\) (Olver 1991).
From the limit in (53) and since the distribution is Gaussian only for p=2 we have from property 4) in Theorem 4 that ϕ_{p}(t) is not infinitely divisible for p≥1 unless p=2.
Another proof that ϕ_{p}(t) is not infinitely divisible for p>2 follows from Theorem 3 since ϕ_{p}(t) has at least one zero, which violates property 1) of Theorem 4. This concludes the proof. □
Next, we show that the Lévy measure in the canonical representation in (51) is an absolutely continuous measure. This also allows us to give a new representation of ϕ_{p}(t) for p∈(0,1] where it is infinitely divisible.
Proposition 14
Proof
See Appendix J. □
Remark 5
6.2 Selfdecomposability of the characteristic function
In this section we are interested in determining whether a GG random variable \(X_{q} \sim \mathcal {N}_{q}(0,\alpha ^{q})\) can be decomposed into a sum of two independent random variables in which one of the random variables is GG. Distributions with such a property are known as selfdecomposable.
Definition 8
where \(Z_{p}\sim \mathcal {N}_{p}(0,1)\) is independent of \( \hat {X}_{\alpha }\).
where \(X_{q} \sim \mathcal {N}_{q}(0,1)\) for every α≥1. The decomposition in (58) finds application in information theory where the existence of the decomposition in (58) guarantees the achievability of Shannon’s bound on the capacity; see (Dytso et al. 2017b) for further details.
is a valid characteristic function.
The expression in (60) is a convex combination of the characteristic function of a point mass at zero and the characteristic function of a Laplace distribution. Therefore, the expression in (60) is a characteristic function.
Checking whether a given function is a valid characteristic function is a notoriously difficult question, as it requires checking whether ϕ_{(q,p,α)}(t) is a positive definite function; see (Ushakov 1999) for an indepth discussion on this topic. However, a partial answer to this question can be given.
Theorem 6

for \((p,q) \in \mathbb {S}_{2}\), ϕ_{(q,p,α)}(t) is a characteristic function (i.e., X_{p} is selfdecomposable for p∈(0,1]∪{2});

for \((p,q) \in \mathbb {R}^{2}_{+} \setminus \mathbb {S}\), ϕ_{(q,p,α)}(t) is not a characteristic function for any α≥1; and

for \( (p,q) \in \mathbb {S}_{1}\) and almost all^{1} α≥1, ϕ_{(q,p,α)}(t) is not a characteristic function.
Proof
See Appendix K. □
For example, when α=2 we have that \(\phi _{(\infty,\infty,\alpha)}(t)=\frac {1}{2} \cos (2t)\), which corresponds to the characteristic function of the random variable \(\hat {X}=\pm 1\) equally likely. Note that in the above example, because zeros of ϕ_{p}(t) occur periodically, we can select α such that the poles and zeros of ϕ_{(q,p,α)}(t) cancel. However, we conjecture that such examples are only possible for p=∞, and for 2<p<∞ zeros of ϕ_{p}(t) do not appear periodically (see Conjecture 1) leading to the following:
Conjecture 2
For 2<q≤p<∞, ϕ_{(q,p,α)}(t) is not a characteristic function for all α>1.
It is not difficult to check, by using the property that convolution with an analytic function is again analytic, that Conjecture 2 is true if p is an even integer and q is any noneven real number.
Discussion and conclusion
In this work we have focused on characterizing properties of the GG distribution. We have shown that for p∈(0,2] the GG random variable can be decomposed into a product of two independent random variables where the first random variable is a positive random variable and the second random variable is also a GG random variable. This decomposition was studied by providing several expressions for the pdf of the positive random variable.
is a proper characteristic function. A partial answer to this question is given next.
Proposition 15

for p>q, is not a valid characteristic function. Therefore, the decomposition in (62) does not exist; and

for p<q, is an integrable function. Moreover, if ϕ_{log(V)}(t) is a valid characteristic function then the pdf of V is given by$$ f_{V}(v)= \frac{1}{2 \pi} \frac{\Gamma \left(\frac{1}{q} \right)}{\Gamma \left(\frac{1}{p} \right)} \int_{\mathbb{R}} v^{it1} \frac{ 2^{\frac{it}{p}} \Gamma \left(\frac{it +1}{p}\right) }{ 2^{\frac{it}{q}} \Gamma \left(\frac{it +1}{q}\right)} dt, \ v>0. $$(63)
Proof
See Appendix L. □
To check if the decomposition in (62) exists for p<q one needs to verify whether the function in (63) is a valid pdf. Because of the complex nature of the integral it is not obvious whether the function in (63) is a valid pdf, and we leave this for future work.
We have also characterized several properties of the characteristic function of the GG distribution such as analyticity, the distribution of zeros, infinite divisibility and selfdecomposability. Moreover, in the regime p∈(0,2) by exploiting the product decomposition we were able to give an exact behavior of the tail of the characteristic function.
We expect that the properties derived in this paper will be useful for a large audience of researchers. For example, in (Dytso et al.2017b,2018) we have used the result in this paper to answer important information theoretic questions about optimal communication over channels with GG noise and optimal compression of GG sources. In view of the fact that GG distributions maximize entropy under L_{p} moment constraints, we also expect that GG distributions will start to play an important role in finding bounds on the entropy of sums of random variables; see for example (Eskenazis et al. 2016) and (Dytso et al. 2017a) where GG distributions are used to derive such bounds.
Appendix A: Proof of Corollary 1
Lemma 1
and let γdenote the Euler’s constant where γ≈0.57721. Then, for every fixed k>0 and log(a)>γ the function g_{k,a}(x) is increasing in x>0.
Proof
Clearly the terms in the summation in (68) are positive under the assumptions of the lemma and, hence, \(\frac {d}{dx} f_{k,a}(x) > 0\). This concludes the proof. □
Observing that \(g_{k}(p)= g_{k,2} \left (\frac {1}{p} \right)\) and log(2)≈0.693>γ≈0.577 concludes the proof that g_{k}(p) is a decreasing function.
The proof is concluded by taking the limit as k→∞ and using that q>p.
Appendix B: Proof of Proposition 3
for p≤q. For completeness the inequality in (69) is shown in Appendix B.1.
where (71) follows from the symmetry and (72) follows from the inequality in (69). This concludes the proof.
B.1 Proof of the inequality in (69)
and then the inequality in (76) follows by the monotonicity of the exponential function. This concludes the proof.
Appendix C: Proof of Theorem 1
The proof is concluded by taking ε small enough and noting that \( \frac { \left (\left (a+1\right)^{p}+a^{p}+1 \right)^{2}}{2} a^{2p} \left (a+1\right)^{2p}1 \ge 0\) for p≤2 and \( \frac { \left (\left (a+1\right)^{p}+a^{p}+1 \right)^{2}}{2} a^{2p} \left (a+1\right)^{2p}1 <0\) for p>2.
An easy way of see that \( \mathrm {e}^{\frac { x^{p}}{ 2 }}\) is a positive definite function is by observing that \( \mathrm {e}^{\frac { x^{p}}{ 2 }}\), for p∈(0,2], is a characteristic function of a stable distribution of order p. The proof then follows by Bochner’s theorem (Ushakov 1999, Theorem 1.3.1.) which guarantees that all characteristic functions are positive definite. For other proofs that \( \mathrm {e}^{\frac { x^{p}}{ 2 }}\) is positive definite for p∈(0,2] we refer the reader to (Lévy 1925) and (Bochner 1937).
Substituting u=x^{2} into (78) completes the proof.
Appendix D: Proof of Proposition 5
where the equalities follow from: a) using the representation of \(\mathrm {e}^{\frac {x^{p}}{ 2 }}\) in Corollary 3; and b) interchanging the order of integration which is justified by Tonelli’s theorem for positive functions.
where the equalities follow from: a) the representation of \(\mathrm {e}^{\frac {x^{p}}{ 2 }}\) in Theorem 1; b) the fact that \(d \nu (t)=\frac {c_{q}}{c_{r}} \frac {1}{t^{\frac {1}{r}}} d\mu _{p}(t)\) is a probability measure; c) because X_{r} is independent of t; and d) renaming \(V_{p,q}= \frac {1}{T^{\frac {1}{r} }}\). Therefore, it follows from (79) that X_{q}=dV_{p,q}·X_{r}.
Appendix E: Proof of Proposition 6
where the (in)equalities follow from: a) using the inequality sin(x)≤1; b) using the power series \(\mathrm {e}^{x}={\sum \nolimits }_{n=0}^{\infty } \frac {x^{n}}{n!}\); and c) using the fact that the integral converges since \( \frac {q}{r}1= \frac {p}{2}1 < 0\) and where we have used that \(p=\frac {2q}{r}\) and p<2 and, hence, \(2^{kq \left (\frac {1}{r} \frac {1}{q} \right)} v^{kq} x^{\frac {kq}{r}} < x\) for large enough x.
where the equalities follow from: a) using the identity \( \sin \left (\frac {\pi k q}{r} \right)= \frac {\mathrm {e}^{\frac {i \pi k q}{r}}\mathrm {e}^{\frac {i \pi k q}{r}} }{2 i} \); b) using the power series expansion \(\mathrm {e}^{x}={\sum \nolimits }_{n=0}^{\infty } \frac {x^{n}}{n!}\); and c) using the identity \(\frac { \mathrm {e}^{ \mathrm {e}^{i \pi x} y}\mathrm {e}^{ \mathrm {e}^{i \pi x} y} }{2i}=\sin \left (\sin \left (\pi x \right) y \right) \mathrm {e}^{ \cos \left (\pi x \right) y}\). Recalling that \(r = \frac {2 q}{p}\) we conclude the proof.
Appendix F: Proof of Proposition 9
The nonnegativity of U_{p}(x) follows from standard trigonometric arguments.
Observe that y_{p}(x)≥0 for x∈(0,1) and all p∈(0,2]. The behavior of h_{p}(x) is slightly more complicated and is given next.
Lemma 2
For p∈(0,1), h_{p}(x)≥0 for all x∈(0,1), and for p∈(1,2]h_{p}(x)≤0 for all x∈(0,1).
Proof
The proof of Lemma 2 is given in Appendix F.1. □
Lemma 2 together with the nonnegativity of y_{p}(x) shows that U_{p}(x) is an increasing function for p∈(0,1) and a decreasing function for p∈(1,2].
Since U_{p}(x) is a strictly monotone function (either decreasing or increasing depending on p), the equation in (86) has only a single solution and therefore g_{p}(x) has only one maximum. Moreover, from (86) the maximum is given by \(\max _{x \in [0,1]} g_{p}(x)= \frac {1}{\mathrm {e} t^{\frac {p}{p1} }}. \) This concludes the proof.
F.1 Proof of Lemma 2
The proof follows by looking at p∈(0,1) and p∈(1,2) separately.
where the inequalities follow from: a) using the fact that \( \cot \left (\frac {\pi p x}{2} \right) >0\) for all x∈(0,1) and all p∈(0,1); b) using the fact that (1−p)^{2}≤1; and c) using the fact that 0<1−p<1 and the fact that \(\tan \left (\frac {\pi (1p) x}{2} \right) \) is a monotonically increasing function for x∈(0,1).
For p∈(1,2) we look at two cases \(x \in (0, \frac {1}{2} ]\) and \(x \in \left (\frac {1}{2}, 1 \right)\). The reason we have to split the domain of x into two parts is because of the \(\cot \left (\frac {\pi p x}{2} \right)\). Note that \(\cot \left (\frac {\pi p x}{2} \right)\ge 0\) for all p∈(1,2) and all \(x \in (0, \frac {1}{2} ]\), but this is not true for the case of \(x \in \left (\frac {1}{2}, 1 \right)\).
Appendix G: Proof of Proposition 10
Therefore, for p=1 we have that \(r=\frac {1}{2}\).
Appendix H: Proof of Theorem 3
The fact that the number of zeros is countable follows from the fact that ϕ_{p}(t) is an analytic function according to Proposition 10. Recall that analytic functions on \(\mathbb {R}\) are either equal to a constant everywhere or have at most countably many zeros; the proof of this fact follows by using the identity theorem and the BolzanoWeierstrass theorem.
For 0<p≤2, the result follows from Theorem 2 since \(\phi _{p}(t) = \mathbb {E} \left [ \mathrm {e}^{\frac {t^{2}V_{G,p}^{2} }{2}}\right ]>0. \) This concludes the proof.
Appendix I: Proof of Proposition 11
Appendix J: Proof of Proposition 14
where g(t) is the cosine transform of the measure G(x).
where (95) follow from Proposition 11.
Appendix K: Proof of Theorem 6
Case of {(p,q):1<p=q}∖{(2,2)}
In this case, since p=q, we return to the proper definition of selfdecomposability (Definition 8). From (Lukacs 1970, Theorem 5.11.1) we have that all distributions with selfdecomposable characteristic functions are infinitely divisible. However, in Theorem 5 we have shown that GG distributions are not infinitely divisible for p∈(1,∞)∖{2}. Therefore, for p∈(1,∞)∖{2} the function ϕ_{(p,p,α)}(t) is not a characteristic function.
Case of {(p,q):0≤p=q≤1}
In this case, since p=q, we return to the proper definition of selfdecomposability (Definition 8). The proof of this case was outlined in (Bondesson 1992, p. 118) and it required the following definitions:
Definition 9
 1(Extended Generalized Gamma Convolution (EGGC) (Bondesson 1992, p.105).) An EGGC is a distribution on \(\mathbb {R}\) such that the bilateral Laplace transform \(\psi (s)=\mathbb {E}[\mathrm {e}^{sX}], \, s\in \mathbb {C}\), defined at least for Re(s)=0, has the form$$ \psi(s)=\mathrm{e}^{bs+\frac{cs^{2}}{2} +\int \left(\log \left(\frac{t}{ts} \right) \frac{st}{1+t^{2}} \right) dU(t) }, $$(97)where \(b\in \mathbb {R}, c \ge 0\), and dU(t) is a nonnegative measure on \(\mathbb {R} \setminus \{0 \}\) such that$$ \int \frac{1}{1+t^{2}} dU(t)<\infty, \text{ and} \int_{t\le 1}  \log\left(t^{2}\right) dU(t)<\infty. $$(98)
 2(\(\mathcal {\beta }\)Class (Bondesson 1992, p. 73).) A pdf f of a nonnegative random variable belongs to the \(\mathcal {\beta }\)Class if f can be written as follows:$$ f(x)=C x^{\beta1}\frac{h_{1}(x)}{h_{2}(x)}, \, x \ge 0, $$(99)where \(\beta \in \mathbb {R}, c \ge 0\) and, for j=1,2,$$ h_{j}(x)=\mathrm{e}^{b_{j} x+ \int \log\left(\frac{y+1}{y+x} \right) d \Gamma_{j}(y)}, \, x \ge 0, $$(100)where b_{j}≥0 and dΓ_{j}(y) is a nonnegative measure on (0,∞) satisfying$$\int \frac{1}{1+y} d\Gamma_{j}(y)<\infty. $$
 3
(Hyperbolic Completely Monotone (HCM) Function (Bondesson 1992, p. 55>).) A function f:(0,∞)↦(0,∞) is called HCM if, for each u>0, the function \(g(w)=\frac {f(uv)}{f \left (\frac {u}{v}\right)}\) is completely monotone as a function of w=v+v^{−1}.
The following results are needed for our proof.
Theorem 7
 1
(Bondesson 1992, p. 107) An EGGC distribution is selfdecomposable.
 2
(Bondesson 1992, Theorem 7.3.3) Let X and Y be two independent random variables such that the distribution of X is EGGC and the distribution of Y is in the βClass. If X is symmetric, then \(\sqrt {Y}X\) has an EGGC distribution.
 3
(Bondesson 1992, Theorem 7.3.4) Let Y be a symmetric random variable on \(\mathbb {R}\) with a pdf f_{Y}. Then \(Y \stackrel {d}{=} \sqrt {V} Z_{2}\) is a Gaussian mixture such that the distribution of V is in the βClass if and only if \(g(t)= f_{Y}(\sqrt {2t})\), t>0, is the Laplace transform of an HCMfunction (or a degenerate function).
 4
(Bosch and Simon 2016) Let f_{α}:(0,∞)↦(0,∞) be a pdf of a positive αstable distribution (i.e., the Laplace transform of f_{α} is equal to \(\mathrm {e}^{t^{\alpha }}\)). Then f_{α} is HCM if and only if \( \alpha \in (0, \frac {1}{2})\).
First observe that the pdf of a GG random variable composed with \(\sqrt {2t}\) is given by \(f_{X_{p}}(\sqrt {2t})= c_{p} \mathrm {e}^{2^{\frac {p}{2}1}t^{\frac {p}{2}}}, \,t >0\), and is a Laplace transform, up to a normalization constant, of an αstable positive random variable (see discussion in “Connection to stable distributions” section).
Next, let g_{p/2}(x),x>0, denote the pdf of an αstable distribution of order \(\frac {p}{2}\). Clearly, g_{p/2}(x) is an inverse Laplace transform of \(f_{X_{p}}(\sqrt {2t})\) up to a normalization constant. Now by Theorem 7 Property 4) we have that g_{p/2}(x) is an HCM function for all \(\frac {p}{2} \in (0, \frac {1}{2}]\). Therefore, \(f_{X_{p}}(\sqrt {2t})\) is a Laplace transform of an HCM function, and by Theorem 7 Property 3) \(f_{X_{p}}\) is a pdf of a Gaussian mixture \(X_{p} \stackrel {d}{=}\sqrt {V} X_{2}\) where the distribution of V is in the βClass. By Theorem 7 Property 2) and Property 1) we have that for all \(\frac {p}{2} \in (0, \frac {1}{2}] X_{p}\) has an EGGC distribution and is selfdecomposable.
Case of q>p>0
where the (in)equalities follow from: a) Jensen’s inequality; and b) the independence of \(\hat {X}_{\alpha }\) and Z_{p}, and that \(\mathbb {E}[\hat {X}_{\alpha }]=0\).
However, by Corollary 1 for p<q we have that \(\alpha \ge {\lim }_{k \to \infty } \left (\frac {\mathbb {E}[ Z_{p}^{k} ] }{\mathbb {E}[X_{q}^{k}] }\right)^{\frac {1}{k}} =\infty ;\) therefore, there exists no α≥1 that can satisfy (102) for all k≥1.
Case of p=2 and q<2
Note that in the case of p=2 and q<2 we want to show that there is no \(\hat {X}_{\alpha }\) such that the convolution leads to \( f_{X_{q}}(y) = c_{2} \mathbb {E} \left [ \mathrm {e}^{\frac {\left (y\hat {X}_{\alpha }\right)^{2}}{2}} \right ]\) where by definition \(f_{X_{q}}(y) = \frac {c_{q}}{\alpha } \mathrm {e}^{\frac {y^{q}}{2 \alpha ^{q}}}\). Such an \(\hat {X}_{\alpha }\) does not exist since the convolution preserves analyticity. In other words, the convolution with an analytic pdf must result in an analytic pdf. Noting that \(f_{X_{q}}(y)\) is not analytic for q<2 (i.e., the derivative at zero is not defined) leads to the desired conclusion.
Case of p>2 and q≤2
Now for p>2 and q≤2 the function ϕ_{(q,p,α)}(t) has a pole but no zeros by Theorem 3. Therefore, for the case of p>2 and q≤2 there exists a t_{0}, namely the pole of ϕ_{(q,p,α)}(t), such that ϕ_{(q,p,α)}(t) is not continuous at t=t_{0}. This violates the condition that the characteristic function is always a continuous function of t and, therefore, ϕ_{(q,p,α)}(t) is not a characteristic function for all α≥1.
Case of p>q>2
For the case of p>q>2 the function \(\phi _{(q,p,\alpha)}(t)=\frac {\phi _{q}(\alpha t)}{\phi _{p}(t)}\) has both poles and zeros by Theorem 3. Moreover, let t_{1} be such that ϕ_{p}(t_{1})=0 and we can always choose an α such that ϕ_{q}(αt_{1})≠0 and ϕ_{(q,p,α)}(t_{1})=∞. In other words, we choose an α such that the poles do not cancel the zeros. Therefore, there exists an α such that ϕ_{(q,p,α)}(t) is not a continuous function of t and therefore is not a characteristic function. Finally, because the number of zeros is at most countable (see Theorem 3) the above argument holds for almost all α≥1.
Case of q<p<2
Finally, for q<p<2 the result follows from Proposition 12 where it is shown that \( {\lim }_{t \to \infty } \phi _{(q,p,\alpha)}(t)=\infty \), which violates the fact that the characteristic function is bounded. This concludes the proof.
Appendix L: Proof of Proposition 15
Declarations
Acknowledgements
The authors would like to thank Professor Alexander Lindner from the Ulm University for providing references (Bondesson 1992) and (Bosch and Simon 2016), which immediately lead to the conclusion that the GG distributions in p∈(0,1] are selfdecomposable.
Funding
The work of A. Dytso and H.V. Poor was supported by the U.S. National Science Foundation under Grant CNS1702808. The work of S. Shamai and R. Bustin was supported by the European Union’s Horizon 2020 Research and Innovation Programme Grant 694630.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables vol. 55. Courier Corporation, Chelmsford (1964).MATHGoogle Scholar
 Algazi, V. R., Lerner, R. M.: Binary detection in white nonGaussian noise. M.I.T. Lincoln Lab. 18(Res. DS2138), 241–250 (1964).Google Scholar
 ArellanoValle, R. B., Richter, W. D.: On skewed continuous ℓ _{n,p}symmetric distributions. Chil. J. Stat. 3(2), 193–212 (2012).MathSciNetGoogle Scholar
 Banerjee, S., Agrawal, M.: Underwater acoustic noise with generalized Gaussian statistics: Effects on error performance. In: Proceedings of OCEANS  Bergen, 2013 MTS/IEEE, pp. 1–8. IEEE, Bergen (2013).Google Scholar
 Beaulieu, N. C., Young, D. J.: Designing timehopping ultrawide bandwidth receivers for multiuser interference environments. Proc. IEEE. 97(2), 255–284 (2009).View ArticleGoogle Scholar
 Bernard, O., D’Hooge, J., Fribouler, D.: Statistical modeling of the radiofrequency signal in echocardiographic images based on generalized Gaussian distribution. In: Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006, pp. 153–156. IEEE, Arlington (2006).Google Scholar
 Bochner, S.: Stable laws of probability and completely monotone functions. Duke Math. J. 3(4), 726–728 (1937).MathSciNetView ArticleGoogle Scholar
 Bondesson, L.: Generalized gamma convolutions and related classes of distributions and densities. Lect. Notes Stat. 76 (1992).Google Scholar
 Bosch, P., Simon, T.: A proof of Bondesson’s conjecture on stable densities. Ark Matematik. 54(1), 31–38 (2016).MathSciNetView ArticleGoogle Scholar
 Cover, T., Thomas, J.: Elements of Information Theory: Second Edition. Wiley, Hoboken (2006).MATHGoogle Scholar
 De Simoni, S.: Su una estensione dello schema delle curve normali di ordine r alle variabili doppie. Statistica. 37, 447–474 (1968).Google Scholar
 de Wouwer, G. V., Scheunders, P., Dyck, D. V.: Statistical texture characterization from discrete wavelet representations. IEEE Trans. Image Process. 8(4), 592–598 (1999).View ArticleGoogle Scholar
 Do, M. N., Vetterli, M.: Waveletbased texture retrieval using generalized Gaussian density and KullbackLeibler distance. IEEE Trans. Image Process. 11(2), 146–158 (2002).MathSciNetView ArticleGoogle Scholar
 Dytso, A., Bustin, R., Poor, H. V., Shamai (Shitz), S.: A view of informationestimation relations in Gaussian networks. Entropy. 19(8), 409 (2017).View ArticleGoogle Scholar
 Dytso, A., Bustin, R., Poor, H. V., Shamai (Shitz), S.: On additive channels with generalized Gaussian noise. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 426–430. IEEE, Aachen (2017).Google Scholar
 Dytso, A., Bustin, R., Tuninetti, D., Devroye, N., Poor, H. V., Shitz, S. S.: On the minimum mean pth error in Gaussian noise channels and its applications. IEEE Trans. Inf. Theory. 64(3), 2012–2037 (2018).MathSciNetView ArticleGoogle Scholar
 Elkies, N., Odlyzko, A., Rush, J.: On the packing densities of superballs and other bodies. Invent. Math. 105(1), 613–639 (1991).MathSciNetView ArticleGoogle Scholar
 Eskenazis, A., Nayar, P., Tkocz, T.: Gaussian mixture entropy and geometric inequalities (2016). Preprint available at https://arxiv.org/abs/1611.04921.
 Fahs, J., AbouFaycal, I.: On properties of the support of capacityachieving distributions for additive noise channel models with input cost constraints. IEEE Trans. Inf. Theory. 64(2), 1178–1198 (2018).MathSciNetView ArticleGoogle Scholar
 Goodman, I. R., Kotz, S.: Multivariate θgeneralized normal distributions. J. Multivar. Anal. 3(2), 204–219 (1973).MathSciNetView ArticleGoogle Scholar
 Gauss, C. F.: Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium vol. 7. Perthes et Besser, Paris (1809).Google Scholar
 GonzalezJimenez, D., PerezGonzalez, F., ComesanaAlfaro, P., PerezFreire, L., AlbaCastro, J. L.: Modeling Gabor coefficients via generalized Gaussian distributions for face recognition. In: Proceedings of the IEEE International Conference on Image Processing, vol. 4, pp. 485–488. IEEE, San Antonio (2007).Google Scholar
 Gupta, A. K., Nagar, D. K.: Matrix Variate Distributions. Chapman and Hall/CRC, London (2018).View ArticleGoogle Scholar
 HoffmanJørgensen, J.: Probability with a View Towards Statistics vol. 2. Routledge, Abingdon (2017).Google Scholar
 Levy, H.: Stochastic dominance and expected utility: survey and analysis. Manag. Sci. 38(4), 555–593 (1992).View ArticleGoogle Scholar
 Lévy, P.: Calcul des Probabilités. GauthierVillars, Paris, France (1925).MATHGoogle Scholar
 Lin, G. D., Huang, J. S.: The cube of a logistic distribution is indeterminate. Aust. J. Stat. 39(3), 247–252 (1997).MathSciNetView ArticleGoogle Scholar
 Lukacs, E.: Characteristic Functions. Griffin, Londong (1970).MATHGoogle Scholar
 Lutwak, E., Yang, D., Zhang, G.: Momententropy inequalities for a random vector. IEEE Trans. Inf. Theory. 53(4), 1603–1607 (2007).MathSciNetView ArticleGoogle Scholar
 Mallat, S. G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Tran. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989).View ArticleGoogle Scholar
 McLachlan, G., Peel, D.: Finite Mixture Models. Wiley, Hoboken (2004).MATHGoogle Scholar
 Miller, J., Thomas, J. B.: Detectors for discretetime signals in nonGaussian noise. IEEE Trans. Inf. Theory. 18(2), 241–250 (1972).View ArticleGoogle Scholar
 Mohamed, O. M. M., JaidaneSaidane, M., Souissi, J.: Modeling of the load duration curve using the asymmetric generalized Gaussian distribution: case of the Tunisian power system. In: Proceedings of the 10th International Conference on Probabilistic Methods Applied to Power Systems, pp. 1–6. IEEE, Rincon (2008).Google Scholar
 Moulin, P., Liu, J.: Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors. IEEE Trans. Inf. Theory. 45(3), 909–919 (1999).MathSciNetView ArticleGoogle Scholar
 Nadarajah, S.: A generalized normal distribution. J. Appl. Stat. 32(7), 685–694 (2005).MathSciNetView ArticleGoogle Scholar
 Nielsen, P. A., B.Thomas, J.: Signal detection in Arctic underice noise. In: Proceedings of the 25th Annual Allerton Conference on Communication, Control, and Computing, pp. 172–177. IEEE, Monticello (1987).Google Scholar
 Nielsen, F., Nock, R.: Maxent upper bounds for the differential entropy of univariate continuous distributions. IEEE Signal Process. Lett. 24(4), 402–406 (2017).View ArticleGoogle Scholar
 Olver, F.: Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22(5), 1460–1474 (1991).MathSciNetView ArticleGoogle Scholar
 Ozarow, L. H., Wyner, A. D.: On the capacity of the Gaussian channel with a finite number of input levels. IEEE Trans. Inf. Theory. 36(6), 1426–1428 (1990).MathSciNetView ArticleGoogle Scholar
 Pogány, T. K., Nadarajah, S.: On the characteristic function of the generalized normal distribution. C. R. Math. 348(3), 203–206 (2010).MathSciNetView ArticleGoogle Scholar
 Poor, H. V., Thomas, J. B.: Locally optimum detection of discretetime stochastic signals in nonGaussian noise. J. Acoust. Soc. Am. 63(1), 75–80 (1978).MathSciNetView ArticleGoogle Scholar
 Poularikas, A. D.: Handbook of Formulas and Tables for Signal Processing. CRC Press, Boca Raton (1998).View ArticleGoogle Scholar
 Richter, W. D.: Generalized spherical and simplicial coordinates. J. Math. Anal. Appl. 336(2), 1187–1202 (2007).MathSciNetView ArticleGoogle Scholar
 Richter, W.D.: Geometric disintegration and starshaped distributions. J. Stat. Distrib. Appl. 1(1), 20 (2014).View ArticleGoogle Scholar
 Richter, W.D.: Exact inference on scaling parameters in norm and antinorm contoured sample distributions. J. Stat. Distrib. Appl. 3(1), 8 (2016).View ArticleGoogle Scholar
 Schilling, R. L., Song, R., Vondracek, Z.: Bernstein Functions: Theory and Applications vol. 37. Walter de Gruyter, Berlin, Germany (2012).View ArticleGoogle Scholar
 Sharifi, K., LeonGarcia, A.: Estimation of shape parameter for generalized Gaussian distributions in subband decompositions of video. IEEE Trans. Circ. Syst. Video Technol. 5(1), 52–56 (1995).View ArticleGoogle Scholar
 Soury, H., Yilmaz, F., Alouini, M. S.: Average bit error probability of binary coherent signaling over generalized fading channels subject to additive generalized Gaussian noise. IEEE Commun. Lett. 16(6), 785–788 (2012).View ArticleGoogle Scholar
 Soury, H., Alouini, M. S.: New results on the sum of two generalized Gaussian random variables. In: Proceedings of the 2015 IEEE Global Conference on Signal and Information Processing, pp. 1017–1021. IEEE, Orlando (2015).Google Scholar
 Subbotin, M.: On the law of frequency of error. Matematicheskii Sb. 31, 296–301 (1923).MATHGoogle Scholar
 Stewart, J.: Positive definite functions and generalizations, an historical survey. Rocky Mt. J. Math. 6(3), 409–434 (1976).MathSciNetView ArticleGoogle Scholar
 Stoyanov, J.: Krein condition in probabilistic moment problems. Bernoulli Journal. 6(5), 939–949 (2000).MathSciNetView ArticleGoogle Scholar
 Ushakov, N. G.: Selected Topics in Characteristic Functions. Walter de Gruyter, Berlin, Germany (1999).View ArticleGoogle Scholar
 van Harn, K., Steutel, F.: Infinite Divisibility of Probability Distributions on the Real Line. Taylor & Francis, New York (2003).MATHGoogle Scholar
 Varanasi, M. K., Aazhang, B.: Parametric generalized Gaussian density estimation. J. Acoust. Soc. Am. 86(4), 1404–1415 (1989).View ArticleGoogle Scholar
 Vasudevay, R., Kumari, J. V.: On general error distributions. ProbStat Forum. 06, 89–95 (2013).MathSciNetGoogle Scholar
 Viswanathan, R., Ansari, A.: Distributed detection of a signal in generalized Gaussian noise. IEEE Trans. Acoust. Speech, Signal Process. 37(5), 775–778 (1989).View ArticleGoogle Scholar
 Westerink, P. H., Biemond, J., Boekee, D. E.: Subband coding of color images. In: Subband Image Coding, pp. 193–227. Springer, Boston (1991).View ArticleGoogle Scholar
 Widder, D. V.: The Laplace Transform. 1946. Princeton University Press, Princeton (1946).Google Scholar
 Zolotarev, V. M.: Onedimensional Stable Distributions vol. 65. American Mathematical Society, Providence (1986).View ArticleGoogle Scholar