# Generalization of the Alpha-Stable Distribution with the Degree of Freedom

Stephen H. Lihn<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: May 12, 2024. v1.0.0512. Dedicated to my mother on the Mother’s Day of
2024. Full Version )

###### Abstract.

A Wright function based framework is proposed to combine and extend several
distribution families. The $\alpha$-stable distribution is generalized by
adding the _degree of freedom_ parameter. The PDF of this two-sided super
distribution family subsumes those of the original $\alpha$-stable, Student’s
t distributions, as well as the exponential power distribution and the
modified Bessel function of the second kind. Its CDF leads to a fractional
extension of the Gauss hypergeometric function. The degree of freedom makes
possible for valid variance, skewness, and kurtosis, just like Student’s t.
The original $\alpha$-stable distribution is viewed as having one _degree of
freedom_ , that explains why it lacks most of the moments. A skew-Gaussian
kernel is derived from the characteristic function of the $\alpha$-stable law,
which maximally preserves the law in the new framework. To facilitate such
framework, the stable count distribution is generalized as the fractional
extension of the generalized gamma distribution. It provides rich
subordination capabilities, one of which is the fractional $\chi$ distribution
that supplies the needed ’degree of freedom’ parameter. Hence, the ”new”
$\alpha$-stable distribution is a ”ratio distribution” of the skew-Gaussian
kernel and the fractional $\chi$ distribution. Mathematically, it is a new
form of higher transcendental function under the Wright function family. Last,
the new univariate symmetric distribution is extended to the multivariate
elliptical distribution successfully.

###### Key words and phrases:

fractionally generalized distribution, the Wright function, fractional chi
distribution, stable count distribution, generalized gamma distribution,
$\alpha$-stable distribution, Student’s t distribution, exponential power
distribution, multivariate distribution, Feller square root process,
fractional hypergeometric function.

###### 2020 Mathematics Subject Classification:

Primary: 33C05, 33C10, 33C15, 33E20, 60E07, 60E05, 60E10, 60G53, 62E15, 62H10,
65C10

Disclaimer: Mr. Lihn is a quant researcher at Atom Investors LP, Austin, TX
78746, USA. This mathematical treatise is not related to the business of Atom
Investors LP. I attest there is no conflict of interest.

##  1\. Introduction

The aim of this work is to propose a two-sided, super distribution family,
called _the generalized $\alpha$-stable distribution_ (GAS), that subsumes
three major distributions, plus one transcendental function:

  * •

the $\alpha$-stable distribution,

  * •

Student’s t distribution,

  * •

the exponential power distribution, and

  * •

the modified Bessel function of the second kind111It is the characteristic
function of Student’s t distribution.

The literature review is as follows.

In 1908, William S. Gosset published a paper on measuring the distribution on
samples of finite size under the pseudonym ”Student”[30]. He pioneered a
distribution later called _Student’s t distribution_ or simply _the t
distribution_ , that plays a crucial role in statistical inferences and
hypothesis testing. The distribution has a major parameter called _degrees of
freedom_ $k$, hence the notation $t_{k}$, with the Cauchy distribution at
$k=1$, and the standard normal distribution $\mathcal{N}=\mathcal{N}(0,1)$ at
$k\to\infty$.

Student’s t distribution can be constructed as the ratio of a standard normal
variable and the square root of a $\chi^{2}$ distribution, divided by degrees
of freedom. That is, $t_{k}\sim\mathcal{N}/\sqrt{\chi_{k}^{2}/k}$. The
$\chi^{2}$ distribution was first derived by F. R. Helmert in 1872 from his
work on least squares adjustment and error propagation[12]. Later in 1900,
Karl Pearson defined the $\chi$ distribution as the square root of a
$\chi^{2}$ variable: $\chi_{k}\sim\sqrt{\chi_{k}^{2}}$.

This history is summarised in our first and central guiding equation:

(1) $\displaystyle t_{k}$
$\displaystyle\sim\mathcal{N}/\,\overline{\chi}_{k},\quad\quad\,\,\text{where}\,\,\overline{\chi}_{k}:=\chi_{k}/\sqrt{k}.$

In 1925, Paul Lévy published his seminal book on the $\alpha$-stable
distribution[13]. The distribution has a major parameter, among others, called
_the stability index_ $\alpha\in(0,2]$. In this work, we call the extension to
the $\alpha$ dimension _fractional_.

The $\alpha$-stable distribution family also contains the Cauchy distribution
at $\alpha=1$, and the normal distribution at $\alpha=2$. Are the two
commonalities between Student’s t and $\alpha$-stable coincident?

A main goal of this paper is to show that this is not a coincidence – There is
a super distribution family that subsumes both distributions seamlessly.
However, its construction requires the fractional extension of the $\chi$
distribution: $\chi_{k}\to\chi_{\alpha,k}$, and a branch of mathematical
functions called _higher transcendental function_.

In 1933, E. M. Wright published his work on the asymptotic theory of
partitions[32, 33] that proposed a particular type of higher transcendental
function, with the notation of $W_{\lambda,\delta}(z)$ where $\lambda,\delta$
are two shape parameters. It was later collected into the Bateman manuscript
in 1955, but tucked under the Chapter of the Mittag-Leffler function (Chapter
18.1 of Vol 3)[2]. In recent decades, this function is found to have an
essential role in the theory of fractional calculus and the $\alpha$-stable
distribution, hence is called _the Wright function_.

In 1952, William Feller pioneered a new stable parameterization
$(\alpha,\theta)$ in his work on the Riesz-Feller fractional derivative[8],
linking the fractional calculus and the $\alpha$-stable distribution
together[20, 26]. It is called _the Feller parameterization_ in this work, in
which $\theta$ is the skewness parameter in the form of a trigonometric angle.
Among many parameterizations in the $\alpha$-stable distribution (sometimes
very confusing), this is the primary parameterization used here (See
Definition 3.7 of [23]). Therefore, the $\alpha$-stable distribution takes the
notation of $L_{\alpha}^{\theta}$.

Later in Feller’s 1971 seminal book on the probability theory[9], the series
representation for the PDF of the $\alpha$-stable distribution was given. When
$\theta=\pm\alpha$, the distribution becomes one-sided (or called extremal).
Its PDF $L_{\alpha}(x)$ $(x>0)$ is based on one of the most important variants
of the Wright function:
$L_{\alpha}(x)=x^{-1}{W}_{{-\alpha},0}\left(-{x^{-\alpha}}\right)$ (See Table
2).

In a nutshell, $\chi_{\alpha,k}$ is a Wright function extension of $\chi_{k}$.
We would properly standardize $\chi_{\alpha,k}$ to
$\overline{\chi}_{\alpha,k}$, then extend $t_{k}$ from (1) to

(2) $\displaystyle L_{\alpha,k}$
$\displaystyle\sim\mathcal{N}/\,\overline{\chi}_{\alpha,k}$

after which the skewness $\theta$ is added, and we obtain the enlarged
distribution family $L_{\alpha,k}^{\theta}$. We make sure when $k=1$, it
conforms to the $\alpha$-stable distribution: $L_{\alpha,1}^{\theta}$ =
$L_{\alpha}^{\theta}$.

We can also look at it from a different angle. We take the $\alpha$-stable
distribution $L_{\alpha}^{\theta}$, add $k$ from $t_{k}$ to it, and form
$L_{\alpha,k}^{\theta}$. We would get to the same place.

Since 1990’s, Mainardi et al. have been exploring extensively the relation
between the Wright function, the fractional calculus, and the $\alpha$-stable
distribution[18, 21, 19]. I consider this work as a continuation of such
exploration.

In 1962, E. W. Stacy proposed a super distribution family called _the
generalized gamma distribution_ (GG). Many one-sided distributions commonly
used for parametric models are part of it, such as the exponential
distribution and the exponential power distribution, the half-normal, Weibull,
Rayleigh, gamma, $\chi$, and $\chi^{2}$ distributions (See Table  ‣ 5.3).

The Wright function and GG have a deep connection. The large-$x$ asymptotic of
a PDF based on the Wright-function becomes a GG-style PDF, apart from a
constant. We can jokingly say _they are joined at the tail_.

One contribution of this work is to have GG fractionally extended, called _the
generalized stable count distribution_ (GSC), which serves as the parent of
the fractional $\chi$ distribution. Obviously, GSC is the largest one-sided
distribution family in this scope. The following is a brief review of my past
effort.

In 2017, I discussed the $\alpha$-stable law with Professor Mulvey at
Princeton University. Later that year, I derived _the stable count
distribution_ $\mathfrak{N}_{\alpha}(x)$ as the conjugate prior of the one-
sided stable distribution $L_{\alpha}(x)$[16, 17, 11, 24].
$\mathfrak{N}_{\alpha}(x)$ measures the distribution of the length $N$ of a
Lévy sum $\sum_{i=1}^{N}X_{i}$ where $X_{i}$’s are i.i.d. stable random
variables. It was found to be expressed elegantly in the Wright function:
$\mathfrak{N}_{\alpha}(x)={\Gamma(\frac{1}{\alpha}+1)}^{-1}{W}_{{-\alpha},0}\left(-{x^{\alpha}}\right)$.

In 2022, I began to organize the subordination structure of the GG-related
distributions via the stable count distribution, which is documented in
Appendix B.1. The GG-style pattern in the PDF of GSC,
$x^{d-1}{W}_{{-\alpha},0}\left(-{x^{p}}\right)$, was discovered (See Table  ‣
5.3).

It turned out such pattern is not a coincidence. GSC becomes GG when
$\alpha\to 0$. GSC is proven to be the fractional extension of GG, solidifying
its theoretical position in the statistical distribution theory. The proof is
based on the nice properties of Mainardi’s auxiliary Wright functions[18].

In 2023, the subordination study was extended to the known symmetric two-sided
distributions, as shown in Appendix B.2. The main structure in the PDF of the
fractional $\chi$ distribution, $\chi_{\alpha,k}(x)\propto
x^{k-2}{W}_{{-\frac{\alpha}{2}},0}\left(-{x^{\alpha}}\right)$, became clear,
when the _continuous Gaussian mixture_ (2) is focused. This is elaborated in
Sections 2 and 7. It combines the $\alpha$ and $k$ parameters into a single
distribution. This is the cornerstone that generalizes the $\alpha$-stable,
Student’s t, and exponential power distributions. All three are combined to
the expanded notation $L_{\alpha,k}^{\theta}$. These are the main
contributions of this paper.

The symmetric case $(\theta=0)$ is very solid and elegant:
$L_{\alpha,k}=L_{\alpha,k}^{0}$. Most of this paper aims to explain this case
in great detail. The skewness generalization is experimental. It is a much
harder task that requires future work. In a tongue-twisting manner, it goes
like:

  * •

$L_{\alpha,1}^{\theta}$: the original $\alpha$-stable distribution,

  * •

$L_{1,k}(k>0)$: Student’s t distribution,

  * •

$L_{\alpha,-1}$: the exponential power distribution.

I also proposed a framework for the generation of the random variables, from
the GSC to $\overline{\chi}_{\alpha,k}$, to $L_{\alpha,k}$. It is based on
Feller’s square-root process in 1951[7]. Hence, this new distributional system
is grounded on an elegant, well known stochastic process. Amazingly, some of
the solutions of the mean-reverting force turn out to be simple polynomials.

Lastly, the univariate Gaussian mixture is extended nicely to the multivariate
distribution. Two kinds are proposed: The first kind is a straightforward copy
from the elliptical distribution. The second kind is an invention that allows
each dimension to have its own shape. The bivariate models from both kinds are
applied to the VIX/SPX222 VIX is the CBOE volatility index; SPX is the S&P 500
index.  data set with success. This is discussed in Section 4.

Figure 1 illustrates the stack of distributions involved in this work. Seven
new distributions are invented, and a handful of existing, classic
distributions are subsumed into them.

Figure 1. The hierarchy of the distributions involved in this work. The Wright
function plays a central role. The green boxes are the contributions from this
work.

The benefits from multiple degrees of freedom, aka $k>1$, are two-fold. First,
the original $\alpha$-stable distribution is viewed as having one degree of
freedom. That explains why it lacks most of the moments. $k>1$ allows the
distribution to have finite moments with increasing $k$: The $n$-th moment
should exist when $k>n$. This addresses, to some extent, the decades-old issue
of non-existing moments in the stable law.

Secondly, for the generalized exponential power distribution, it allows the
slope of the PDF to be continuous everywhere. This deserves some explanations.

In 2017, I incorporated the exponential power distribution,
$\mathcal{E}_{\alpha}(x)=e^{-{|x|}^{\alpha}}/2\Gamma\left(\frac{1}{\alpha}+1\right)$,
into the Hidden Markov Model (HMM) for regime identification[15]. It is
especially important for the crash regime, such that the model can capture the
leptokurtic nature of the financial time series.

A main deficiency of its high-kurtosis distribution is that, when $\alpha\leq
1$, the slope of the PDF is discontinuous at $x\to 0^{\pm}$. A simple example
is $e^{-|x|}$. This appears to violate the intuition that the low-volatility
returns should distribute like normal. This deficiency is also attributed as
having one degree of freedom. It is addressed by the addition of the $k$
dimension: $\mathcal{E}_{\alpha}\to\mathcal{E}_{\alpha,k}$. When $k>1$, the
slope of the PDF is continuous everywhere.

To showcase the new GAS distribution, it is used to fit the daily log returns
of SPX from 1927 to 2020 in Figure 2. This is a very difficult data set to fit
for any existing two-sided distribution. In the _pyro project_[3], it is used
as a failed example that this data set can not be fitted by a single marginal
distribution333 See https://pyro.ai/examples/stable.html.

Nonetheless, GAS produced a good fit. It fits the shape of the histogram
reasonably good in both the density and log-density scales, to the $10^{-3}$
level, which is about 6 standard deviations. It produced the desired excess
kurtosis ($\sim$19) and standardized peak density ($\sim$0.72), and added a
small skewness. Most interestingly, it provides a set of interpretable
parameters that we are familiar with in the Student’s t and $\alpha$-stable
context: $\alpha=0.813,k=3.292$ for the shape, and $\theta=0.08$ for the
skewness. More discussion is in Section 3.4.

Figure 2. Empirical distribution of S&P 500 daily log returns. Data is
standardized and centered for demonstration purpose. The return distribution
is normalized to one standard deviation with zero mean. Fit by GSaS (orange
line) and GAS (red line). The fits are reasonably good up to 6 standard
deviations.

##  2\. Framework of Continuous Gaussian Mixture

The bulk of this paper is focused on the construction of a symmetric two-sided
distribution in the form of a continuous Gaussian mixture. Both the ratio and
product distribution methods are used. In the case of the symmetric
$\alpha$-stable distribution (SaS)[5], the exponential power distribution
comes from its characteristic function (CF). We would like to present a
unified framework and familiarize the reader with the notations, which would
be otherwise subtle and confusing.

Assume the PDF of a two-sided symmetric distribution is $L(x)$ where
$x\in\mathbb{R}$. It has zero mean, $\mathbb{E}(X|L)=0$. Assume the PDF of a
one-sided distribution is $\chi(x)\,(x>0)$ such that

(3) $\displaystyle L(x)$
$\displaystyle:=\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\chi(s)$

This is nothing but the definition of a ratio distribution with a standard
normal variable $\mathcal{N}$. This is the first form of the Gaussian mixture:
$L\sim\mathcal{N}/\chi$.

It also has the equivalent expression in terms of a product distribution by
way of _the inverse distribution_ $\chi^{\dagger}$ such that
$L\sim\mathcal{N}\chi^{\dagger}$. This is the second form of the Gaussian
mixture.

$\chi^{\dagger}$ is closer to our typical understanding of the marginal
distribution of a volatility process. For example, the Brownian motion process
$dX_{t}=\sigma_{t}\,dW_{t}$, measured in a particular time interval $\Delta
t$, we have $\Delta X_{t}\sim L$ and $\sigma_{t}\sim\chi^{\dagger}$.

However, $\chi$ in the first form is more natural in the expression of the
$\alpha$-stable distribution. So we are more inclined to use the ratio
distribution. The reader should keep this subtlety in mind.

######  Lemma 2.1.

(Inverse distribution) The inverse distribution is defined as[10]

(4) $\displaystyle\chi^{\dagger}(s)$
$\displaystyle:=s^{-2}\,\chi\left(\frac{1}{s}\right)$

such that

(5) $\displaystyle\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\chi(s)$
$\displaystyle=\int_{0}^{\infty}\,\frac{ds}{s}\,\mathcal{N}\left(\frac{x}{s}\right)\,\chi^{\dagger}(s)$
$\displaystyle(x\in\mathbb{R})$ (6)
$\displaystyle\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\chi^{\dagger}(s)$
$\displaystyle=\int_{0}^{\infty}\,\frac{ds}{s}\,\mathcal{N}\left(\frac{x}{s}\right)\,\chi(s)$

The proof is straightforward by a change of variable $t=1/s$. You can move
between LHS and RHS easily.

We use the notation $\text{CF}\\{g\\}(t)=\mathbb{E}(e^{itX}|g)$ to represent
the characteristic function transform of the PDF $g(x)$. Note that
$\mathcal{N}$ has a special property that its CF is still itself:
$\text{CF}\\{\mathcal{N}\\}(t)=\sqrt{2\pi}\,\mathcal{N}(t)$.

######  Lemma 2.2.

(Characteristic function transform of $L$) Let $\phi(t)$ be the CF of $L$ such
that
$\phi(t):=\text{CF}\\{L\\}(t)=\int_{-\infty}^{\infty}dx\,\exp(itx)\,L(x)$. (3)
is transformed to

(7) $\displaystyle\phi(t)$
$\displaystyle=\sqrt{2\pi}\,\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{t}{s}\right)\,\chi(s)$
$\displaystyle(t\in\mathbb{R})$

This allows us to define a new distribution pair: $L_{\phi}$ and
$\chi_{\phi}^{\dagger}$, in terms of a product distribution such that

(8) $\displaystyle L_{\phi}(x)$
$\displaystyle:=\int_{0}^{\infty}\,\frac{ds}{s}\,\mathcal{N}\left(\frac{x}{s}\right)\,\chi_{\phi}^{\dagger}(s)$
$\displaystyle(x\in\mathbb{R})$ (9) $\displaystyle\chi_{\phi}^{\dagger}(s)$
$\displaystyle:=\frac{s\,\chi(s)}{\mathbb{E}(X|\chi)}$

where $\mathbb{E}(X|\chi)$ is the first moment of $\chi$. Here
$\chi_{\phi}^{\dagger}$ is the inverse distribution of $\chi_{\phi}$, which
can be reverse-engineered according to (4),

(10) $\displaystyle\chi_{\phi}(s)$
$\displaystyle:=\frac{s^{-3}}{\mathbb{E}(X|\chi)}\,\chi\left(\frac{1}{s}\right)$

We are in an interesting place: We start with an one-sided distribution
$\chi$, we derive two variants from it: $\chi_{\phi}$ and
$\chi_{\phi}^{\dagger}$. We also obtain two two-sided distributions: $L$ and
$L_{\phi}$.

We shall call $\chi_{\phi}$ _the characteristic distribution_ of $\chi$ since
it facilitates the following parallel relation:

$\displaystyle L$ $\displaystyle\sim\mathcal{N}/\chi$ $\displaystyle L_{\phi}$
$\displaystyle\sim\mathcal{N}/\chi_{\phi}$

$\chi$ symbolizes the fractional $\chi$ distribution we are about to
introduce. The $\phi$ suffix will be replaced with the _negation_ (sign
change) of the degree of freedom.

###  2.1. List of Symbols

The next tables describe the notations and symbols that will be used in this
paper. Note that GSC, FCM, GAS, GSaS are new notations. Others are known.

The PDF of Simple Distribution

$\Gamma(x;s)$ | The gamma distribution (used only as a notation)
---|---
${\chi}_{k}(x),{\chi}^{2}_{k}(x)$ | The chi and chi-square distributions
${L}(x)$ | The two-sided Laplace distribution
${\mathcal{N}}(x),\mathcal{N}(x;\mu,\sigma^{2})$ | The normal distribution, ${\mathcal{N}}(x)\sim{\mathcal{N}}(x;0,1)$
$\mathcal{E}_{\alpha}(x)$ | The two-sided exponential power distribution ($e^{-{|x|}^{\alpha}}/2\Gamma\left(\frac{1}{\alpha}+1\right)$)
$\mathfrak{N}_{\alpha}(\nu)$ | The stable count distribution (SC)
$V_{\alpha}(x)$ | The stable vol distribution (SV)
${L}_{\alpha}(x)$ | The one-sided stable distribution, aka $L_{\alpha}^{\alpha}(x)$
$P_{\alpha}({x})$ | The symmetric $\alpha$-stable distribution, aka $L_{\alpha}^{0}(x)$ (SaS)
${t}_{k}(x)$ | Student’s t distribution
${\text{Wb}}(x;k)$ | Weibull distribution
${\text{IG}}(x;k)$ | The inverse gamma distribution (See scipy.stats, but use $a=k$)
${\text{IWb}}(x;k)$ | The inverse Weibull distribution (See scipy.stats, but use $c=k$)

The PDF of Complex Distribution

$f_{\text{\tiny GG}}(x;a,d,p)$ | The generalized gamma distribution (GG)
---|---
$\text{GenGamma}(x;s,c)$ | The generalized gamma distribution too $(c=p,sc=d)$
$L_{\alpha}^{\theta}({x})$ | The $\alpha$-stable distribution
$\mathfrak{N}_{\alpha}(x;\sigma,d,p)$ | The generalized stable count distribution (GSC, new)
$L_{\alpha,k}^{\theta}({x})$ | The generalized $\alpha$-stable distribution (GAS, new)
$L_{\alpha,k}({x})$ | The generalized symmetric $\alpha$-stable distribution (GSaS, new)
$\chi_{\alpha,k}(x)$ | The fractional $\chi$ distribution (new)
$\overline{\chi}_{\alpha,k}({x})$ | The fractional $\chi$-mean distribution (FCM, new)
$\mathcal{E}_{\alpha,k}(x)$ | The generalized exponential power distribution (GExpPow, new)

Other Symbols

$\alpha$ | The shape parameter associated with the Lev́y stability index
---|---
$C$ | The normalization constant for $\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
$d$ | The degree of freedom parameter in $\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
$\theta$ | The skewness parameter in $L_{\alpha,k}^{\theta}({x})$ and $g_{\alpha}^{\theta}(x,s)$
$e^{-z^{\alpha}}$ | The one-sided stretched exponential function
$E_{\alpha}(z)$ | The Mittag-Leffler function
$k$ | The degree of freedom parameter in $L_{\alpha,k}^{\theta}({x}),{t}_{k}(x),\overline{\chi}_{\alpha,k}({x})$
$\mathbb{E}(X^{n}|\text{Dist})$ | The $n$-th moment of the distribution ”Dist”
$m_{n}$ | The $n$-th moment in a local context
$p$ | The shape parameter in $\mathfrak{N}_{\alpha}(x;\sigma,d,p)$, unless mentioned otherwise
$\Gamma(z)$ | The gamma function
$\Gamma(s,z)$ | The upper incomplete gamma function
$\gamma(s,z)$ | The lower incomplete gamma function
$g_{\alpha}^{\theta}(x,s)$ | The skew-Gaussian kernel
$W_{\lambda,\delta}(z)$ | The Wright function (two-parameter)
$W\left[\begin{matrix}a,&b\\\ \lambda,&\mu\end{matrix}\right](z)$ | The four-parameter Wright function

##  3\. Main Results

We will go over the main results in this section, and leave the explanations
and proofs for later sections. The new symmetric $\alpha$-stable distribution
$L_{\alpha,k}$ is constructed in three steps. Then its CF is used to extend
the exponential power distribution. The last part is to add the skewness
$\theta$ experimentally to the new $\alpha$-stable distribution
$L_{\alpha,k}^{\theta}$.

###  3.1. Recap of the Wright Function

The mathematical foundation of this work is the Wright function[32, 33] as
shown in the center of Figure 1. Its series representation is

(11) $\displaystyle W_{\lambda,\delta}(z)$
$\displaystyle:=\sum_{n=0}^{\infty}\frac{z^{n}}{n!\,\Gamma(\lambda n+\delta)}$
$\displaystyle(\lambda\geq-1,z\in\mathbb{C})$

Four $(\lambda,\delta)$ pairs are used extensively in this work. The first
group of the two are $F_{\alpha}(z):=W_{-\alpha,0}(-z)$ and
$M_{\alpha}(z):=W_{-\alpha,1-\alpha}(-z)=F_{\alpha}(z)/(\alpha z)$, where
$\alpha\in[0,1]$. They are used in various definitions and proofs, such as in
(14), (22), (128).

In particular, $M_{\alpha}(z)$ is called _the M-Wright function_ or simply
_the Mainardi function_ (See Appendix C.1)[18, 21, 19]. Conceptually, the
_fractional extension_ of a classic, exponential-based distribution hinges on
two important properties: $M_{0}(z)=\exp(-z)$ and
$M_{\frac{1}{2}}(z)=\frac{1}{\sqrt{\pi}}\exp(-z^{2}/4)$, that lead to (21).

The second group of the two are $W_{-\alpha,-1}(-z)$ and
$-W_{-\alpha,1-2\alpha}(-z)$. Their usefulness is discovered by me for the
generation of random variables, such as in (101) and (133). They are
associated with the derivatives of $F_{\alpha}(z)$ and $M_{\alpha}(z)$. In
some cases, they lead to beautiful polynomial solutions. Section 11 is
dedicated to this subject.

Let’s begin with the 2017 discovery of the stable count (SC) distribution
$\mathfrak{N}_{\alpha}(x)$[16]. Its PDF was first formulated as the conjugate
prior of the one-sided stable distribution $L_{\alpha}(x)$. Soon afterwards,
it was linked to the Wright function:

(12) $\displaystyle\mathfrak{N}_{\alpha}(x)$
$\displaystyle:=\frac{1}{\Gamma(\frac{1}{\alpha}+1)}\frac{1}{x}L_{\alpha}\left(\frac{1}{x}\right)=\frac{1}{\Gamma(\frac{1}{\alpha}+1)}W_{-\alpha,0}(-x^{\alpha})$
$\displaystyle(0<\alpha\leq 1)$

In 2020, a variant of SC was proposed, called the stable vol (SV) distribution
$V_{\alpha}(x)$[17], that works better with a normal variable:

(13) $\displaystyle V_{\alpha}(x)$
$\displaystyle:=\frac{\sqrt{2\pi}\,\Gamma(\frac{2}{\alpha}+1)}{\Gamma(\frac{1}{\alpha}+1)}\,\mathfrak{N}_{\frac{\alpha}{2}}(2x^{2})=\frac{\sqrt{2\pi}}{\Gamma(\frac{1}{\alpha}+1)}\,W_{-\frac{\alpha}{2},0}\left(-{(\sqrt{2}x)}^{\alpha}\right)$
$\displaystyle(0<\alpha\leq 2)$

As the readers can observe, $F_{\alpha}(z)$, $M_{\alpha}(z)$,
$\mathfrak{N}_{\alpha}(x)$, $V_{\alpha}(x)$, $L_{\alpha}(x)$ all point to a
GG-style pattern that subsumes a large number of one-sided distributions. The
pattern is as follows.

###  3.2. New One-Sided Distributions

######  Definition 3.1 (Generalized stable count distribution (GSC)).

GSC is a four-parameter one-sided distribution family, whose PDF is defined as

(14) $\displaystyle\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
$\displaystyle:=C\,{\left(\frac{x}{\sigma}\right)}^{d-1}\,{W}_{{-\alpha},0}\left(-{{\left(\frac{x}{\sigma}\right)}^{p}}\right)$
$\displaystyle(x\geq 0)$

where $\alpha\in[0,1]$ controls the shape of the Wright function; $\sigma$ is
the scale parameter; $p$ is also the shape parameter controlling the tail
behavior $(p\neq 0,dp\geq 0)$; $d$ is the _degree of freedom_ parameter. When
$\alpha\to 1$, the PDF becomes a Dirac delta function: $\delta(x-\sigma)$
assuming $\sigma$ is finite. When $d\geq 1$, all the moments of GSC exist and
have closed forms. See Section 5 for more detail.

The normalization constant $C$ will be derived in Section 5.2 and is shown
below:

(17) $\displaystyle C$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{|p|}{\sigma}\frac{\Gamma(\frac{\alpha
d}{p})}{\Gamma(\frac{d}{p})}&,\,\,\text{for}\,\,\alpha\neq 0,d\neq 0.\\\
\frac{|p|}{\sigma\alpha}&,\,\text{for}\,\,\alpha\neq 0,d=0.\end{array}\right.$

The original stable count distribution (12) is simply
$\mathfrak{N}_{\alpha}(x)=\mathfrak{N}_{\alpha}(x;\sigma=1,d=1,p=\alpha)$. See
Section 5.7.

It is important to note that $d$ and $p$ are allowed to be negative, as long
as $dp\geq 0$. Several examples are found in Table  ‣ 5.3 and Table 2. This
feature is used to define the characteristic distribution of FCM below.

Since the Wright function extends an exponential function to the fractional
space, GSC is the fractional extension of the generalized gamma (GG)
distribution[29], whose PDF is defined as:

(18) $f_{\text{\tiny
GG}}(x;a,d,p)=\frac{|p|}{a\Gamma(\frac{d}{p})}{\left(\frac{x}{a}\right)}^{d-1}e^{-(x/a)^{p}}.$

The parallel use of parameters is obvious, except that $a$ in GG is replaced
with $\sigma$ in GSC to avoid confusion with $\alpha$.

GG is subsumed to GSC in two ways:

(21) $\displaystyle f_{\text{\tiny GG}}(x;\sigma,d,p)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\mathfrak{N}_{0}(x;\sigma,d=d-p,p)&,\,\,\text{at}\,\,\alpha=0.\\\
\mathfrak{N}_{\frac{1}{2}}\left(x;\sigma=\frac{\sigma}{2^{2/p}},d=d-\frac{p}{2},p=\frac{p}{2}\right)&,\,\,\text{at}\,\,\alpha=\frac{1}{2}.\end{array}\right.$

The first line is treated as the definition of GSC at $\alpha=0$. See Sections
5.3 and 5.4 for more detail.

Although the first line is more obvious, it is the second line that leads to
the fractional extension of the $\chi$ distribution.

The readers are reminded that ${W}_{{-\alpha},0}\left(-{x}\right)$ used here
is fully supported by existing software packages. This is due to the fact that
it can be converted to $L_{\alpha}(x)$ by
$L_{\alpha}(x)=x^{-1}{W}_{{-\alpha},0}\left(-{x^{-\alpha}}\right)$ from (12).
And $L_{\alpha}(x)$ can be computed via scipy.stats.levy_stable using
1-Parameterization with beta=1, scale= $\cos(\alpha\pi/2)^{1/\alpha}$ for
$0<\alpha<1$. ($L_{1}(x)=\delta(x-1)$ can’t be computed.)444 See Chapter 1 of
[23] for more detail on different parameterizations. We wouldn’t go into the
issue of stable parameterizations. It might seem somewhat peculiar that we can
use the existing implementation of $L_{\alpha}(x)$ to develop all the new
distributions for proof of concept.

When working with the stable law, $\alpha$ in GSC may become $\alpha/2$ ,
which produces the more recognizable range of $\alpha\in[0,2]$ as in the next
definition.

Let $\chi_{k}$ be the $\chi$ distribution of $k$ _degrees of freedom_ , then
Student’s t distribution $t_{k}$ can be constructed by
$t_{k}\sim\mathcal{N}/(\chi_{k}/\sqrt{k})$[28]. We prefer to standardize it by
dividing out the $\sqrt{k}$ term. So we use a new convention of a bar over
$\chi$ such that $\overline{\chi}_{k}:=\chi_{k}/\sqrt{k}$.

######  Definition 3.2 (Fractional $\chi$-mean distribution (FCM)).

FCM $\overline{\chi}_{\alpha,k}$ is the fractional extension of
$\overline{\chi}_{k}$. It is the main workhorse in this paper. It is a two-
parameter sub-family of GSC, whose PDF for $k>0$ is:

(22) $\displaystyle\overline{\chi}_{\alpha,k}({x})$
$\displaystyle:=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma={\sigma}_{\alpha,k},d=k-1,p=\alpha)$
$\displaystyle(x\geq 0,k>0)$
$\displaystyle\quad\quad\,\,\text{where}\,\,{\sigma}_{\alpha,k}=\frac{{|k|}^{1/2-1/\alpha}}{\sqrt{2}}$
$\displaystyle=\frac{\alpha\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\,{\left(\frac{1}{{\sigma}_{\alpha,k}}\right)}^{k-1}{x}^{k-2}\,W_{\frac{\alpha}{2},0}\left(-{\left(\frac{x}{{\sigma}_{\alpha,k}}\right)}^{\alpha}\right)$

and $\alpha\in[0,2]$ matches exactly Lévy’s stability index in the stable law.
See Lemma 3.3 when $k=1$.

The characteristic FCM ($\chi_{\phi}$ in Section 2) is defined in the negative
$k$ space, whose PDF is:

(23) $\displaystyle\overline{\chi}_{\alpha,-k}(x)$
$\displaystyle:=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma=({\sigma}_{\alpha,k})^{-1},d=-k,p=-\alpha)$
$\displaystyle(x\geq 0,k>0)$

This is used to subsume the exponential power distribution to GSaS below.
Negative $k$ is not in the scope until (36).

To use a single expression for both positive and negative $k$’s, let
$h(k)=(1+\operatorname{sgn}(k))/2$ be the Heaviside (step) function. The PDF
can be consolidated to:

(24) $\displaystyle\overline{\chi}_{\alpha,k}(x)$
$\displaystyle=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma={({\sigma}_{\alpha,k})}^{\operatorname{sgn}(k)},d=k-h(k),p=\operatorname{sgn}(k)\,\alpha)$
$\displaystyle(x\geq 0,k\in\mathbb{R}_{\neq 0})$

This looks cumbersome, but it can save some duplication in a few places.

######  Lemma 3.1.

(FCM Reflection Formula) Assume $k>0$, FCM has the reflection formula that
resembles (10):

(25) $\displaystyle\overline{\chi}_{\alpha,-k}(x)$
$\displaystyle=\frac{1}{x^{3}\,\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\,\overline{\chi}_{\alpha,k}\left(\frac{1}{x}\right)$
$\displaystyle\,\,\text{where}\,\,\mathbb{E}(X|\overline{\chi}_{\alpha,k})={\sigma}_{\alpha,k}\,\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{k}{\alpha})}{\Gamma(\frac{k}{2})}\,\,\text{(See
\eqref{eq:fcm-moment})}$

It is equivalent to the following relation in terms of the moments:

(26) $\displaystyle\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,-k})$
$\displaystyle=\frac{\mathbb{E}(X^{-n+1}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})},\,\,k>0.$

In particular, when $n=1$,
$\mathbb{E}(X|\overline{\chi}_{\alpha,-k})=1/\mathbb{E}(X|\overline{\chi}_{\alpha,k})$.
The first moment is reciprocal.

When $k$ is positive, $k$’s meaning is exactly the same as that of Student’s
t, since $\overline{\chi}_{1,k}=\chi_{k}/\sqrt{k}$, which can be proven easily
from (21). On the other hand, $\overline{\chi}_{\alpha,1}$ is called
_fractional chi-1_. It is used to construct the SaS distribution:
$P_{\alpha}:=L_{\alpha}^{0}\sim\mathcal{N}/\overline{\chi}_{\alpha,1}$. This
important point will be elaborated in Lemma 3.3.

The parameter space outside $(\alpha=1\,|\,k=1)$ is our innovation - a main
contribution of this paper. See Section 7 for more detail.

######  Lemma 3.2.

(FCM Asymptotics) As $k\to\pm\infty$, FCM becomes a delta function at its
asymptotic mean. For $k\gg 1$,

(27) $\displaystyle\overline{\chi}_{\alpha,k}(x)$
$\displaystyle\sim\delta(x-m_{\alpha}),\,\,\,\,\text{where}\,\,m_{\alpha}={\alpha}^{-1/\alpha}.$

On the other hand, $\overline{\chi}_{\alpha,-k}(x)\sim\delta(x-1/m_{\alpha})$.

This property ensures that the two-sided distributions constructed from FCM
become a normal distribution at $k\to\pm\infty$. Hence, this lemma is
associated with the Central Limit Theorem.

Note that $m_{1}=1$ is preserved for Student’s t; $m_{2}=1/\sqrt{2}$ is
preserved for $\alpha$-stable.

The speed of approaching a delta function is determined by how fast the
variance decreases:
$\mathbb{E}(X^{2}|\overline{\chi}_{\alpha,k})-\mathbb{E}(X|\overline{\chi}_{\alpha,k})^{2}\to
0$. It is faster for larger $\alpha$, and can be very slow for small $\alpha$
below 1.

######  Definition 3.3.

(Inverse FCM) The inverse distribution of FCM can be constructed by ”inverting
$\sigma$ and negating $d$ and $p$” so to speak. The results are simple due to
the FCM reflection rule:

(28) $\displaystyle\overline{\chi}_{\alpha,k}^{\dagger}(x)$
$\displaystyle:=\frac{x\,\overline{\chi}_{\alpha,-k}(x)}{\mathbb{E}(X|\overline{\chi}_{\alpha,-k})}=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma=({\sigma}_{\alpha,k})^{-1},d=-(k-1),p=-\alpha)$
$\displaystyle(k>0)$ (29)
$\displaystyle\overline{\chi}_{\alpha,-k}^{\dagger}(x)$
$\displaystyle:=\frac{x\,\overline{\chi}_{\alpha,k}(x)}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma={\sigma}_{\alpha,k},d=k,p=\alpha)$

Note that (29) is similar to (22) except $d$ is incremented by one. And (28)
is similar to (23) except $d$ is incremented by one.

We are jumping ahead to emphasize that the inverse FCM allows the generalized
exponential power distribution $\mathcal{E}_{\alpha,k}$ in (34) defined as
$\mathcal{N}/\,\overline{\chi}_{\alpha,-k}^{\dagger}$ to be mapped to the
negative $k$ space of GSaS as
$\mathcal{N}/\,\overline{\chi}_{\alpha,-k}^{\dagger}=\mathcal{N}\,\overline{\chi}_{\alpha,-k}=L_{\alpha,-k}$.

As a validation, $\overline{\chi}_{\alpha,-1}^{\dagger}(x)$ is equal to the
stable vol distribution $V_{\alpha}(x)$ in (13), as expected by its original
design (explained in Section 5.7).

In Figure 3, we plot the FCM distribution that came out of the S&P 500’s GSaS
fit, which is a distribution of the inverse of volatility.

Figure 3. The FCM distribution from the S&P 500’s GSaS fit. Its scale has been
standardized because the variance of the log-return distribution was
normalized to one. This is a distribution of the inverse of volatility. The
smaller $x$ the more volatile the market was.

All aspects of FCM are well defined up to this point. Next, we are going to
take apart the $\alpha$-stable distribution.

######  Definition 3.4 (The skew-Gaussian kernel).

The skew-Gaussian kernel $g_{\alpha}^{\theta}(x,s)$ is derived from the
$\alpha$-stable law.

The minus-log of the stable CF is defined in Feller’s parameterization[9] as
$\psi_{\alpha}^{\theta}(\zeta):=\exp\left(\text{sgn}(\zeta)\frac{i\theta\pi}{2}\right){|\zeta|}^{\alpha}$,
where $\theta$ is the skewness parameter confined by the so-called _Feller-
Takayasu diamond_ : $\theta\leq\min\\{\alpha,2-\alpha\\}$ and
$\alpha\in(0,2]$[20, 19].

The stable PDF is
$L_{\alpha}^{\theta}({x}):=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\zeta\,\exp(-\psi_{\alpha}^{\theta}(\zeta))\,e^{-ix\zeta}$.
We want it to be decomposed according to
$L_{\alpha}^{\theta}({x})=\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\overline{\chi}_{\alpha,1}(s)$.
The solution of $g_{\alpha}^{\theta}(x,s)$ is

(30) $\displaystyle g_{\alpha}^{\theta}(x,s)$
$\displaystyle=\frac{1}{q\pi}\int_{0}^{\infty}dt\,\cos\left(\tau\,{(st)}^{\alpha}+\frac{x}{q}\,st\right)e^{-t^{2}/2}$
$\displaystyle(x\in\mathbb{R},s\geq 0)$
$\displaystyle\,\,\text{where}\,\,q=\cos(\theta\pi/2)^{1/\alpha},\,\,\tau=\tan(\theta\pi/2),\,\,\theta\neq
1.$

Feller’s parameterization provides a clean environment to dissect the
subordination structure in the stable CF. See Section 6 for more detail.

In general, $g_{\alpha}^{\theta}(x,s)$ is not a distribution, since some parts
of the function can be negative. But it satisfies
$\int_{-\infty}^{\infty}g_{\alpha}^{\theta}(x,s)dx=1/s$. One exception is
$g_{\alpha}^{0}(x,s)=\mathcal{N}(xs)$. Also note that
$g_{\alpha}^{\theta}(x,0)=1/q\sqrt{2\pi}$, independent of $x$.

Most of the difficulty in computing the $\alpha$-stable PDF arises from
integrating some kind of highly oscillatory functions. See Section 3 of [23]
for a general discussion.

In our approach, that difficulty is concentrated in $g_{\alpha}^{\theta}(x,s)$
for a non-zero $\theta$. The range of the integral is confined by
$e^{-t^{2}/2}$, which is nice. But when the oscillation frequency is high, it
gets harder to integrate with the ordinary quad utility in the scipy package.
We switch to the quadosc function in the mpmath package[22] with more success
in accuracy, but some performance is sacrificed.

######  Lemma 3.3.

Fractional Chi-1. When the skewness is absent: $\theta=0$, we get
$q=1,\tau=0$, there is no ”complicated oscillation” from the cosine function.
$g_{\alpha}^{0}(x,s)=\mathcal{N}(xs)$ greatly simplifies the matter. This
leads to the SaS distribution as
$P_{\alpha}:=L_{\alpha}^{0}\sim\mathcal{N}/\overline{\chi}_{\alpha,1}$. That
is, all the shape information in SaS is determined by
$\overline{\chi}_{\alpha,1}$.

The $\overline{\chi}_{\alpha,1}$ term is called _fractional chi-1_ , which is
the fractional version of the $\chi$ distribution of one degree of freedom
from the absolute of a random variable defined as such. This is a new concept,
used as a building block for $\overline{\chi}_{\alpha,k}$.

Here we briefly explain how to go from $\overline{\chi}_{\alpha,1}$ to
$\chi_{\alpha,k}$. See Section 7.3 in full detail.

$\overline{\chi}_{\alpha,1}$ is
$\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma=\frac{1}{\sqrt{2}},d=0,p=\alpha)$,
whose PDF is

(31) $\displaystyle\overline{\chi}_{\alpha,1}(x)$
$\displaystyle=2x^{-1}\,{W}_{{-\frac{\alpha}{2}},0}\left(-{{(\sqrt{2}x)}^{\alpha}}\right)$

$\overline{\chi}_{\alpha,1}$ has some peculiar behaviors. For $\alpha<1$,
$\overline{\chi}_{\alpha,1}(x)$ diverges as $x^{\alpha-1}$ for small $x$. And

$\displaystyle\lim_{\alpha\to 0}\overline{\chi}_{\alpha,1}(x)$
$\displaystyle\approx\alpha/(ex)$
$\displaystyle(x>\alpha\,\,\text{approximately})$
$\displaystyle\overline{\chi}_{1,1}(x)$
$\displaystyle=\sqrt{2/\pi}\,e^{-x^{2}/2}$
$\displaystyle\overline{\chi}_{2,1}(x)$ $\displaystyle=\delta(x-1/\sqrt{2})$

The first line implies $\displaystyle\lim_{\alpha\to
0}P_{\alpha}(x)\approx\overline{\chi}_{\alpha,1}(x)/2\approx\alpha/(2ex)$, for
not too small $x$, e.g. $x>\alpha/2$.

The $x^{k-2}$ Term. In the $k>0$ case, the PDF $\overline{\chi}_{\alpha,k}(x)$
comes from $\overline{\chi}_{\alpha,1}(x)$ multiplied by the surface area of
the $(k-1)$-dimensional sphere, which is $\propto x^{k-1}$. Note that
$\overline{\chi}_{\alpha,1}(x)\propto
x^{-1}{W}_{{-\frac{\alpha}{2}},0}\left(-{{\left(2x^{2}\right)}^{\alpha/2}}\right)$.
Hence, $\chi_{\alpha,k}(x)\propto
x^{k-2}{W}_{{-\frac{\alpha}{2}},0}\left(-{{\left(2x^{2}\right)}^{\alpha/2}}\right)$.
This is the origin of the crucial $x^{k-2}$ term.

Once $\chi_{\alpha,k}$ is defined, it is standardized to FCM such that
$\overline{\chi}_{\alpha,k}:=\chi_{\alpha,k}/{\sigma}_{\alpha,k}$. FCM is used
to construct all the subsequent two-sided distributions.

###  3.3. New Two-Sided Distributions

We focus on the symmetric case where $\theta=0$ first, and defer the skew case
$\theta\neq 0$ to the latter.

When $\theta=0$, we have $g_{\alpha}^{0}(x,s)=\mathcal{N}(xs)$, independent of
$\alpha$. The new generalized symmetric distribution is solely based on FCM
that subsumes the SaS, Student’s t, and exponential power distributions.

######  Definition 3.5.

(Generalized symmetric $\alpha$-stable distribution (GSaS)) GSaS is elegantly
constructed as a ratio distribution:
$L_{\alpha,k}\sim\mathcal{N}/\,\overline{\chi}_{\alpha,k}$. Hence, its PDF is

(32) $\displaystyle L_{\alpha,k}({x})$
$\displaystyle:=\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle(x\in\mathbb{R},\alpha\in(0,2],k\in\mathbb{R}_{\neq 0})$

It follows naturally that $L_{\alpha,1}=P_{\alpha}$ for SaS, and
$L_{1,k}=t_{k}$ for Student’s t when $k>0$. On the other hand,
$L_{\alpha,-1}=\mathcal{E}_{\alpha}$ for the exponential power.

######  Lemma 3.4.

(GSaS Version of Central Limit Theorem) When $|k|\gg 1$, it is straightforward
from Lemma 3.2 that $L_{\alpha,k}$ becomes a normal distribution. This
statement is similar to the Central Limit Theorem (CLM). The variance
asymptotics of $L_{\alpha,k}$ is $(\alpha^{2/\alpha})^{\operatorname{sgn}(k)}$
at $\alpha\to 2$ or $|k|\to\infty$. This is consistent with both
$\alpha$-stable at $\alpha=2$ and Student’s t at $\alpha=1$ by design.

Its CF becomes a product distribution, according to (7), such as

(33) $\displaystyle\phi_{\alpha,k}(\zeta):=\text{CF}\\{L_{\alpha,k}\\}(\zeta)$
$\displaystyle=\sqrt{2\pi}\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{\zeta}{s}\right)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle(\zeta\in\mathbb{R})$

For $k>0$, it is an expanded form of the inverse lambda decomposition (59). It
describes, in the $\theta=0$ case, how the $\alpha$-stable law is modified
from $\exp(-{|\zeta|}^{\alpha})$ at $k=1$ to a generalized form of $k>1$.

Since $L_{\alpha,k}({x})$ with $k>0$ covers Student’s t as well,
mathematically speaking, $\phi_{\alpha,k}(\zeta)$ forms a high transcendental
function that subsumes both the stretched exponential function and the
modified Bessel function of the second kind (due to
$\text{CF}\\{t_{k}\\}(\zeta)$). See Section 8.6 for more detail.

######  Definition 3.6 (Generalized exponential power distribution (GEP)).

According to Lemma 2.2, $\phi_{\alpha,k}(\zeta)$ can be treated as a
distribution. In this case, its $k=1$ base is the exponential power
distribution whose PDF is
$\mathcal{E}_{\alpha}(x)=e^{-{|x|}^{\alpha}}/2\Gamma\left(\frac{1}{\alpha}+1\right)$[14,
25]. We take it to the next level:

(34) $\displaystyle\mathcal{E}_{\alpha,k}(x)$
$\displaystyle:=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\,\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{x}{s}\right)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle(x\in\mathbb{R},k>0)$ (35)
$\displaystyle=\int_{0}^{\infty}\,\frac{ds}{s}\,\mathcal{N}\left(\frac{x}{s}\right)\,\overline{\chi}_{\alpha,-k}^{\dagger}(s)$

where the normalization constant $\mathbb{E}(X|\overline{\chi}_{\alpha,k})$ is
FCM’s first moment in (71). And $\mathcal{E}_{\alpha,1}=\mathcal{E}_{\alpha}$.

Furthermore, according to Lemma 2.1, (35) is inverted to a ratio distribution
form:

(36) $\displaystyle\mathcal{E}_{\alpha,k}(x)$
$\displaystyle=\int_{0}^{\infty}s\,ds\,\mathcal{N}\left(xs\right)\,\overline{\chi}_{\alpha,-k}(s)$
$\displaystyle=L_{\alpha,-k}(x)$

Therefore, GSaS subsumes GEP beautifully when the negative $k$ domain is
defined properly in FCM.

The $k$ dimension mitigates a flaw in $\mathcal{E}_{\alpha}(x)$: when
$\alpha\leq 1$, the slope of the PDF is discontinuous at $x=0$. That is,
$\lim_{x\to 0^{-}}\frac{d}{dx}\,\mathcal{E}_{\alpha}(x)$ $\neq\lim_{x\to
0^{+}}\frac{d}{dx}\,\mathcal{E}_{\alpha}(x)$. Now as long as $k>1$,
$\frac{d}{dx}\,\mathcal{E}_{\alpha,k}(x)$ is continuous everywhere. See
Section 9 for more detail.

Since $\mathcal{N}(xs)$ in (32) doesn’t take either $\alpha$ or $k$. All the
shape information comes from our innovation, $\overline{\chi}_{\alpha,k}(s)$.
For instance, the peak PDF $L_{\alpha,k}(0)$ is the first moment of
$\overline{\chi}_{\alpha,k}$.

Excess Kurtosis. The moments of GSaS have closed form solutions. Of particular
interest is the excess kurtosis, plotted in Figure 4 in the $(k,\alpha)$
coordinate. Notice that a major division occurs along the line of
$k=5-\alpha$. In the region where $0<k\leq 5-\alpha$, there are complicated
patterns caused by the infinities of the gamma function. Only small pockets of
valid kurtosis exist.

######  Lemma 3.5.

In the region where $k>5-\alpha$, the excess kurtosis is a continuous function
with positive values. At large $k$’s, the closed form of the moments can be
expanded by Sterling’s formula. The excess kurtosis (exKurt) becomes part of a
linear equation:

(37)
$\displaystyle{\left(s-\frac{1}{2}\right)}=\left(\frac{k-3}{4}\right)\,\log\left(1+\frac{\text{exKurt}}{3}\right),\quad\,\,\text{where}\,\,\,s=1/\alpha$

This equation shows how GSaS works under the Central Limit Theorem. GSaS
approaches a normal distribution: $L_{\alpha,k}\to\mathcal{N}$ when the excess
kurtosis becomes zero. This can happen from two directions: when $\alpha\to
2$, or when $k\to\infty$.

The contour plot of excess kurtosis is shown in the $(k,s)$ coordinate in
Figure 5. It is visually amusing. Notice the singular point at $s=1/2,k=3$.

The excess kurtosis of $\mathcal{E}_{\alpha,k}$ has a closed form too:

(38) $\displaystyle\text{exKurt}(\mathcal{E}_{\alpha,k})$
$\displaystyle=\frac{3\,\mathbb{E}(X|\overline{\chi}_{\alpha,k})\,\mathbb{E}(X^{5}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X^{3}|\overline{\chi}_{\alpha,k})^{2}}-3$

which is shown in Figure 6. The linear contour levels are clear indication
that the distribution has an orderly structure in terms of kurtosis.

Figure 4. The contour plot of excess kurtosis in GSaS by $(k,\alpha)$.  Figure
5. The contour plot of excess kurtosis in GSaS by $(k,s)$ where $s=1/\alpha$,
in order to show the linear equation described in (37) for large $k$’s. Figure
6. The contour plot of excess kurtosis (38) of the generalized exponential
power distribution $\mathcal{E}_{\alpha,k}$ by $(k,s)$ where $s=1/\alpha$. The
linear contour levels can be clearly observed. We allow the formula to be
evaluated down to $k=-2$ to show the singular point at $s=1/2,k=-2$.

######  Definition 3.7 (Generalized $\alpha$-stable distribution (GAS)).

GAS is an experimental skew distribution based on GSaS. The GAS
$L_{\alpha,k}^{\theta}$ is a ratio such that $L_{\alpha,k}^{\theta}\sim
g_{\alpha}^{\theta}/\overline{\chi}_{\alpha,k}$. Hence, its PDF is

(39) $\displaystyle L_{\alpha,k}^{\theta}({x})$
$\displaystyle:=\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle(x\in\mathbb{R})$

However, the PDF of a distribution must be positive-definite (Bochner’s
Theorem). This narrows the range of $\theta$ as $k$ increases.

GAS adds the $k$ dimension to $\alpha$-stable by expanding from
$\overline{\chi}_{\alpha,1}$ to $\overline{\chi}_{\alpha,k}$ in the integral.
It subsumes $\alpha$-stable as $L_{\alpha}^{\theta}\sim
L_{\alpha,1}^{\theta}$, and subsumes Student’s t as $t_{k}\sim L_{1,k}^{0}$.
See Section 10 for more detail.

GAS admits valid variance, skewness, and kurtosis, _as long as there are a few
degrees of freedom_ , just like you would expect in Student’s t. (See Lemma
3.5 about the validity of kurtosis.) It inherits the skewness from the
$\alpha$-stable law - a property that very few two-sided skew distributions
have, if any.

Most real-world data analyses begin with the summary statistics of the first
four moments. Plus a few more, such as the peak PDF, and the 95% and 99%
confidence levels. GAS can address all of them. This is a large improvement to
$\alpha$-stable.

###  3.4. Explanation of the S&P 500 Fit

Next, we explain how we fitted the S&P 500 daily log return data that produced
Figure 2. We used GSaS in the first pass. The values of $\alpha$ and $k$ are
obtained such that the GSaS agree with the empirical kurtosis (20) and
standardized peak density (0.71). In the second pass, a small skewness is
added to GAS and we fine-tuned the fit.

However, the solution for the S&P 500 fit falls in an unconventional region as
shown in Figure 7. It is in a small pocket near $\alpha=1,k=3$.
Conventionally, as we know from Student’s t, the kurtosis only exists for $k$
greater than 4. But in our new $(\alpha,k)$ space, it is possible for $k$ to
be less than 4 with valid kurtosis, although only in small pockets. We found
the solution by tracing the contours of the peak PDF and excess kurtosis as
shown in Figure 8. (The peak PDF has a closed form solution and is more stable
than the kurtosis numerically.) The intersection near $\alpha=0.81,k=3.3$ is
the solution that satisfies both properties. This is the only solution for the
S&P 500 data in the low degrees of freedom region where $k<5$.

Figure 7. The region near $(\alpha=1,k=3)$ that contains the S&P 500 solution.
Figure 8. The S&P 500 solution is found by tracing the contours of the excess
kurtosis (blue line) and the peak PDF (red line). The intersection is the
solution that satisfies both properties.

##  4\. Multivariate Distribution

The appearance of $\mathcal{N}(xs)$ makes it possible to extend GSaS to an
$n$-dimensional multivariate symmetric distribution with the concept of the
elliptical distribution[31]. We propose two viable options. And we use the
daily return distributions of the S&P 500 Index (SPX) and CBOE VIX Index (VIX)
since 1990 to examine the bivariate use case.

We need to upgrade $\mathcal{N}(xs)$ to a multivariate normal distribution
properly. To set it up, we vectorize $x$ to $\mathbf{x}\in\mathbb{R}^{n}$, and
introduce the covariance matrix $\mathbf{\Sigma}\in\mathbb{R}^{n\times n}$,
which is invertible to $\mathbf{\Sigma}^{-1}$. Its determinant is
$|\mathbf{\Sigma}|$. There are two routes to go from here.

###  4.1. Multivariate of the First Kind - Elliptical

The first route is simpler and produces very meaningful results. All
dimensions share the same shape from a single FCM. So $(xs)^{2}$ in
$\mathcal{N}(xs)$ is replaced to
$\mathbf{x}^{\intercal}\mathbf{\Sigma}^{-1}\mathbf{x}\,s^{2}$. Such elliptical
distribution closely resembles that of a multivariate normal. The ellipse
structure $\mathbf{x}^{\intercal}\mathbf{\Sigma}^{-1}\mathbf{x}$ is perserved.

We use an extended notation
$\mathcal{N}\left(\mathbf{x};\mathbf{\mu},\mathbf{\Sigma}\right)$ to represent
the PDF of a multivariate normal distribution. But for the purpose of
exploring fundamental properties, we don’t want to be bothered with the
location parameter. Therefore, we set its $\mathbf{\mu}=0$.

######  Definition 4.1.

(GSaS Elliptical Multivariate Distribution) The PDF of the first kind is

(40) $\displaystyle L_{\alpha,k}(\mathbf{x};\mathbf{\Sigma})$
$\displaystyle:=\int_{0}^{\infty}\,ds\,\,\overline{\chi}_{\alpha,k}(s)\,\mathcal{N}\left(\mathbf{x};0,\frac{\mathbf{\Sigma}}{s^{2}}\right)$
(41)
$\displaystyle=\frac{1}{(2\pi)^{n/2}{|\mathbf{\Sigma}|}^{1/2}}\int_{0}^{\infty}s^{n}\,ds\,\,\overline{\chi}_{\alpha,k}(s)\left[\exp\left(-\frac{s^{2}}{2}\,\mathbf{x}^{\intercal}\mathbf{\Sigma}^{-1}\mathbf{x}\right)\right]$

The scipy.stats.multivariate_normal function is used in our prototype
implementation. It is obvious from the first line that the total density is
integrated to 1.

When $\alpha=1,k>0$, $L_{1,k}(\mathbf{x};\mathbf{\Sigma})$ becomes the
multivariate t distribution. This is validated numerically with
scipy.stats.multivariate_t. On the other hand, due to Lemma 3.4,

$\displaystyle\lim_{\alpha\to
2\,\text{or}\,k\to\infty}L_{\alpha,k}\left(\mathbf{x};{\alpha^{-2\alpha}}\,\mathbf{\Sigma}\right)=\mathcal{N}\left(\mathbf{x};0,\mathbf{\Sigma}\right)$

These two validations are critically _important_. They show that the
univariate GSaS extends naturally to multivariate through the combination of
the elliptical distribution and FCM.

From (41), its peak PDF is

(42) $\displaystyle L_{\alpha,k}(\mathbf{0};\mathbf{\Sigma})$
$\displaystyle=\frac{\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,k})}{(2\pi)^{n/2}{|\mathbf{\Sigma}|}^{1/2}}$

Its variance (aka the second moment) is

(43) $\displaystyle\mathbb{E}(X_{i}X_{j})$
$\displaystyle=\mathbb{E}(X^{-2}|\overline{\chi}_{\alpha,k})\,\mathbf{\Sigma}_{i,j}$

Its $i$-th dimension marginal PDF is a GSaS:

(44) $\displaystyle L_{\alpha,k}^{(i)}(x;\mathbf{\Sigma})$
$\displaystyle=\frac{1}{\sigma_{i}}L_{\alpha,k}\left(\frac{x}{\sigma_{i}}\right),\quad\,\,\text{where}\,\,\sigma_{i}=\sqrt{\mathbf{\Sigma}_{i,i}}$

###### Proof.

The proof of (44) is straightforward. The marginal distribution of a
multivariate normal distribution is the down-sized multivariate normal
distribution with the covariance matrix from those selected dimensions. See
Section 3.3.1 of [31]. ∎

Hence, the $i$-th marginal variance is
$\mathbb{E}(X^{2}|L_{\alpha,k})\,\mathbf{\Sigma}_{i,i}$, the same as
$\mathbb{E}(X_{i}^{2})$ in (43) since
$\mathbb{E}(X^{2}|L_{\alpha,k})=\mathbb{E}(X^{-2}|\overline{\chi}_{\alpha,k})$
from Lemma 8.1. They match nicely.

The marginal peak density is
$L_{\alpha,k}^{(i)}(0;\mathbf{\Sigma})=\frac{1}{\sqrt{2\pi}\sigma_{i}}\mathbb{E}(X|\overline{\chi}_{\alpha,k})$.

The multivariate extension of GEP (aka $k<0$) has not been validated since
there is no implementation in the scipy package. But we do find it useful as
an alternative model in Section 4.3.

###  4.2. Multivariate of the Second Kind - Adaptive

The second route is more complex, but also more adaptive for practical use.
For instance, in a multi-asset scenario, it allows each asset to have its own
shape according to its marginal distribution. The downside is that the
elliptic structure is distorted.

In the previous method, we learn the advantage of using multivariate normal to
construct our multivariate PDF. The $\mathcal{N}\left(\mathbf{x};...\right)$
term can be integrated away easily to make sure the total density is equal to
1. We follow the same methodology but expand on the FCM structure.

For simplicity, we use $\Theta_{i}$ to represent the $i$-th parameterization:
$(\alpha_{i},k_{i})$. And $\Theta$ is the collection of all $\Theta_{i}$’s. We
vectorize $s$ to $\mathbf{s}\in\mathbb{R}^{n}$. Each $s_{i}$ comes from a
distinct $\overline{\chi}_{\Theta_{i}}$ in that dimension.

The covariance matrix $\mathbf{\Sigma}$ is sandwiched by $1/\mathbf{s}$ such
that the covariance matrix becomes
$(\mathbf{s}^{\dagger})^{\intercal}\,\mathbf{\Sigma}\,\mathbf{s}^{\dagger}$
where $\mathbf{s}^{\dagger}=\operatorname{diag}(\mathbf{s})^{-1}$. It is then
enveloped by the integrals of the multi-dimensional FCMs. To put it together -

######  Definition 4.2.

(GSaS Adaptive Multivariate Distribution) The PDF of the second kind is

(45) $\displaystyle L_{\Theta}(\mathbf{x};\mathbf{\Sigma})$
$\displaystyle:=\prod_{i}\int_{0}^{\infty}\,ds_{i}\,\,\overline{\chi}_{\Theta_{i}}(s_{i})\,\mathcal{N}\left(\mathbf{x};0,(\mathbf{s}^{\dagger})^{\intercal}\,\mathbf{\Sigma}\,\mathbf{s}^{\dagger}\right),\quad\,\,\text{where}\,\,\mathbf{s}^{\dagger}=\operatorname{diag}(\mathbf{s})^{-1}$

This object is far more computationally intense than
$L_{\alpha,k}(\mathbf{x};\mathbf{\Sigma})$ due to the multi-dimensional
integral on $\mathbf{s}$.

Note that, in the case of identity $\mathbf{\Sigma}$,
$L_{\Theta}(\mathbf{x};\mathbf{\Sigma})$ does factorize into the product of
the marginal one-dimensional distributions. This is a positive feature.
However, this PDF retains the circular contour only in the center region. The
contour could be distorted into rectangle-barbell shapes with round edges in
the tail regions.

Its peak PDF is

(46) $\displaystyle L_{\Theta}(\mathbf{0};\mathbf{\Sigma})$
$\displaystyle=\frac{\prod_{i}\mathbb{E}(X|\overline{\chi}_{\Theta_{i}})}{(2\pi)^{n/2}{|\mathbf{\Sigma}|}^{1/2}}$

Its variance (aka the second moment) is

(49) $\displaystyle\mathbb{E}(X_{i}X_{j})$
$\displaystyle=\left\\{\begin{array}[]{ll}\mathbb{E}(X^{-2}|\overline{\chi}_{\Theta_{i}})\,\mathbf{\Sigma}_{i,i}&,\,\text{for}\,\,i=j.\\\
\mathbb{E}(X^{-1}|\overline{\chi}_{\Theta_{i}})\,\mathbb{E}(X^{-1}|\overline{\chi}_{\Theta_{j}})\,\mathbf{\Sigma}_{i,j}&,\,\text{for}\,\,i\neq
j.\end{array}\right.$

Its $i$-th dimension marginal PDF is a GSaS:

(50) $\displaystyle L_{\Theta}^{(i)}(x;\mathbf{\Sigma})$
$\displaystyle=\frac{1}{\sigma_{i}}L_{\alpha_{i},k_{i}}\left(\frac{x}{\sigma_{i}}\right),\quad\,\,\text{where}\,\,\sigma_{i}=\sqrt{\mathbf{\Sigma}_{i,i}}$

Such marginal distribution can adapt to each sample’s shape. This is the major
improvement over (44). The $i$-th marginal variance is
$\mathbb{E}(X^{2}|L_{\alpha,k})\,\mathbf{\Sigma}_{i,i}=\mathbb{E}(X_{i}^{2})$
in (49). They match nicely.

The $i$-th marginal peak density is
$L_{\Theta}^{(i)}(0;\mathbf{\Sigma})=\frac{1}{\sqrt{2\pi}\sigma_{i}}\mathbb{E}(X|\overline{\chi}_{\alpha_{i},k_{i}})$.
Hence, the peak density ratio between marginal and multivariate is

$\displaystyle\frac{\prod_{i}L_{\Theta}^{(i)}(0;\mathbf{\Sigma})}{L_{\Theta}(\mathbf{0};\mathbf{\Sigma})}$
$\displaystyle=\left[\frac{|\mathbf{\Sigma}|}{\prod_{i}\mathbf{\Sigma}_{i,i}}\right]^{1/2}$

In the bivariate case, this ratio becomes $\sqrt{1-\rho^{2}}$ where $\rho$ is
the correlation and $|\rho|\leq 1$. On one hand, when $\rho=0$, this ratio is
1. On the other hand, the closer $\rho$ is to 1, the smaller this ratio is.
Conversely, the ratio from the sample data implies the model correlation.

In conclusion, the differences between the elliptical version and the adaptive
version are (a) how the peak density is formed; (b) the shapes of the marginal
distributions; and (c) the off-diagonal elements of the variance.

###  4.3. Bivariate Application - SPX and VIX

In this section, we use the bivariate distributions to model the joint daily
return distributions of the S&P 500 Index (SPX) and CBOE VIX Index (VIX) from
1/1990 to 3/2024. We use this case to demonstrate the fitting procedure and
visualize the features mentioned above. This is the pinnacle of this work.

First, the sample statistics are calculated from the full sample. The sample
covariance matrix is $\mathbf{\Sigma}=$ [[ 0.00486232, -0.00055669],
[-0.00055669, 0.0001304]]. The correlation is $\rho=-0.70$. The excess
kurtoses are 15.8 for VIX, and 17.4 for SPX.

Second, we fit the marginal distributions of VIX and SPX with two GSaS’s. The
result is shown in Figure 9. To manage a reasonable size of the bins in the
histograms, the data is truncated at [-0.3, 0.3] for VIX, and [-0.05, 0.05]
for SPX. 200 bins are used. Each GSaS is rescaled to the sample’s standard
deviation: $\sqrt{\mathbf{\Sigma}_{0,0}}$ and $\sqrt{\mathbf{\Sigma}_{1,1}}$.
The model mean is the same as the sample mean in the plots. For demonstration
purpose, the means in the bivariate models are discarded.

Each dimension has its own shapes: $\alpha_{0}=0.64,k_{0}=5.50$ for VIX; and
$\alpha_{1}=0.88,k_{1}=3.20$ for SPX. The parameters are chosen such that the
peak density and the excess kurtosis between the model and sample are matched
reasonably well.’ Skewness is not handled.

The same search procedure shown in Figure 8 is applied to each marginal
distribution.

According to Figure 4, The high sample kurtosis in conjunction with $\alpha$
slightly below 1 indicates $k$ in the neighborhood of 5. This observation is
true for VIX, but not for SPX.

SPX distribution has a sharper peak. The height exceeds the range that any
$k\geq 4$ can afford. The solution is still located in the pocket near
$\alpha\approx 1$ and $k\approx 3$, as shown in Figures 7 and 8. This is a
peculiar feature of SPX index, regardless much shorter history in this case.

To configure the bivariate model distributions, we designed an adjustment
procedure for the model $\mathbf{\Sigma}$. The main obstacle is that, if we
simply feed the parameters obtained above to the bivariate models, the peak
joint density will come out lower than the sample’s peak density, which is
1186, printed in the upper right plot of Figure 10.

We attribute the shortage to insufficient correlation. The adjustment is
treated as an optimization problem: The free variable is an adjustment factor.
The model variance of each dimension is scaled down by the factor, while the
absolute of the correlation is scaled up by the factor. Construct a new model
covariance matrix. The objective is to increase the model’s peak joint density
such that its percentage deviation to the sample peak density is less than the
adjustment factor.

For the adaptive bivariate model, $\rho$ is moved to -0.81. For the elliptical
bivariate model, $\rho$ is moved to -0.79. Since the latter only takes one set
of the shape parameter, we use the simple averages from the marginals:
$\langle\alpha\rangle=(\alpha_{0}+\alpha_{1})/2=0.76$,
$\langle{k}\rangle=(k_{0}+k_{1})/2=4.35$.

The model outputs are shown in the lower two contour plots in Figure 10. The
results are satisfactory. The tilts of the contours are in agreement with that
of the sample (Upper right). The adaptive contour is particularly interesting,
or even a bit exaggerating. It has a rectangle-barbell shape, with four
corners sticking out, which looks more like what’s from the data. Elliptical
contour appears to be too simple and naive.

Figure 9. The The marginal distributions of VIX and SPX daily returns from
1/1990 to 3/2024. Each is fitted with a GSaS by matching their peak densities,
standard deviations, excess kurtosis, and shapes of the distribution. Each
GSaS is shifted to match the mean of the data. The y-axes of the left plots
are in the log scale.  Figure 10. The bivariate distribution of VIX and SPX
daily returns from 1/1990 to 3/2024. _Upper left_ : The scatter plot of the
data points, from which the sample covariance matrix and correlation are
calculated. _Upper right_ : The contour plot from the 2D histogram of the
data. 200 bins in each dimension. This shows approximately what kind of
contours should be expected. They are more like rectangles than ellipses.
_Lower left_ : The contour plot from the elliptical bivariate model
distribution. The tilt of the ellipses agrees with the data (about
$45^{\circ}$ in the chart). _Lower right_ : The contour plot from the adaptive
bivariate model distribution. Its shape is like rectangle-barbell shape, with
four corners sticking out, which looks more like what’s in the raw data.

##  5\. GSC: Generalized Stable Count Distribution

###  5.1. GSC PDF and Explanation

We recap the PDF of GSC (14) from Definition 3.1 and compare it with that of
GG (18)555 The CDF of a GG is a regularized incomplete gamma function
$\Gamma(\cdot,\cdot)$. Hence, it can also be parameterised such as
$\text{GenGamma}(\frac{x}{a};s=\frac{d}{p},c=p):=\Gamma\left(s,{\left(\frac{x}{a}\right)}^{c}\right)$.
side by side:

$\displaystyle\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
$\displaystyle:=C\quad\quad{\left(\frac{x}{\sigma}\right)}^{d-1}\,W_{-\alpha,0}\left(-{\left(\frac{x}{\sigma}\right)}^{p}\right)$
$\displaystyle f_{\text{\tiny GG}}(x;a,d,p)$
$\displaystyle:=\frac{|p|}{a\Gamma(\frac{d}{p})}{\left(\frac{x}{a}\right)}^{d-1}\,\,e^{-(x/a)^{p}}$

The parallel in notations between the two PDFs should be obvious.

GSC is the fractional extension of GG. The exponential function in GG is
_replaced_ with the Wright function: $e^{-z}\rightarrow W_{-\alpha,0}(-z)$.
The additional parameter $\alpha\in[0,1]$ controls the shape of the Wright
function. The scale parameter is changed from $a$ to $\sigma$ to avoid
confusion between $a$ in GG and $\alpha$ in GSC.

The other two parameters $d,p$ are kept the same intentionally: $d$ is the
_degree of freedom_ parameter; $p$ is the second shape parameter controlling
the tail behavior $(p\neq 0,dp\geq 0)$. In our use cases, $p$ is either
$\alpha$ or $2\alpha$. In the later case, $p$ is exactly the stability index.

The range of $\alpha$ is constrained to $[0,1]$ due to the analytic property
of $W_{-\alpha,0}(-z)$. When $\alpha\to 1$, it becomes a Dirac delta function
$\delta(z-1)$. This is a well known result.

On the other hand, when $\alpha\to 0$, GSC becomes GG. See Section 5.3 for
proof. A broad class of distributions belong to GG, see Table  ‣ 5.3.

###  5.2. Determination of C

We derive the value of the normalization constant $C$ in (17) as following.

###### Proof.

The normalization constant $C$ in (14) is obtained from the requirement that
the integral of the PDF must be 1:

$\displaystyle\int_{0}^{\infty}\,N_{\alpha}(x;\sigma,d,p)\,dx$
$\displaystyle=\,\frac{C\sigma}{|p|}\frac{\Gamma(\frac{d}{p})}{\Gamma(\frac{d}{p}\alpha)}=1$

where the integral is carried out by the moment formula of the Wright function
(123).

Hence, $C$ is determined as

$\displaystyle C$
$\displaystyle=\frac{|p|}{\sigma}\frac{\Gamma(\frac{d}{p}\alpha)}{\Gamma(\frac{d}{p})}$
$\displaystyle(\alpha\neq 0,d\neq 0)$

We typically constrain $dp\geq 0$ and $p$ is typically positive. But it
becomes negative in the inverse distribution and/or characteristic
distribution types. So we need $|p|$ to ensure $C$ is positive.

For the case of $\alpha\neq 0$ and $d\to 0$, due to (137), we have

$\displaystyle C$ $\displaystyle=\frac{|p|}{\sigma\alpha}$
$\displaystyle(\alpha\neq 0,d=0)$

These two cases are combined to form (17). ∎

###  5.3. Mapping between GSC and GG at alpha = 0

All of the gamma-based, exponential-based distributions can be expressed by
GSC at $\alpha=0$. We prove the first line of (21) as following.

###### Proof.

For certain proofs, it is more convenient to express the PDF of GSC in terms
of the M-Wright function $M_{\alpha}(z)$ in (128):

(51) $\displaystyle\mathfrak{N}_{\alpha}(x;\sigma,d,p)$ $\displaystyle=\alpha
C\,{\left(\frac{x}{\sigma}\right)}^{d+p-1}\,M_{\alpha}\left({\left(\frac{x}{\sigma}\right)}^{p}\right)$

We apply the property $M_{0}(z)=\exp(-z)$ mentioned below (130) to
$\mathfrak{N}_{\alpha}(x;\sigma,d,p)$ from (51), and obtain:

$\displaystyle\lim_{\alpha\to 0}\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
$\displaystyle=\lim_{\alpha\to 0}\alpha
C\,{\left(\frac{x}{\sigma}\right)}^{d+p-1}\,\exp\left(-{\left(\frac{x}{\sigma}\right)}^{p}\right)$
$\displaystyle=\left(\lim_{\alpha\to
0}\alpha\Gamma\left(\frac{d}{p}\alpha\right)\right)\frac{p/\sigma}{\Gamma(\frac{d}{p})}\,{\left(\frac{x}{\sigma}\right)}^{d+p-1}\,\exp\left(-{\left(\frac{x}{\sigma}\right)}^{p}\right)$
$\displaystyle=\frac{p/\sigma}{\Gamma(\frac{d+p}{p})}\,{\left(\frac{x}{\sigma}\right)}^{d+p-1}\,\exp\left(-{\left(\frac{x}{\sigma}\right)}^{p}\right)$
$\displaystyle=f_{\text{\tiny GG}}(x;\sigma,d+p,p)$

Hence, we reach the mapping of GG to GSC at $\alpha=0$ described in the first
line of (21):

(52) $\displaystyle f_{\text{\tiny GG}}(x;\sigma,d,p)$
$\displaystyle=\mathfrak{N}_{0}(x;\sigma,d-p,p)$

∎

The GSC mapping of classic distributions at $\alpha=0$ is illustrated in Table
‣ 5.3.

|  |  GG: $f_{\text{\tiny GG}}(x;\sigma,d,p)$ |  GSC: $\mathfrak{N}_{0}(x;\sigma,d,p)$
---|---|---|---
Distribution (PDF) | Classic Equiv. | $\sigma$ | $d$ | $p$ | $\alpha$ | $\sigma$ | $d$ | $p$
Stretched Exp: $\mathcal{E}_{\alpha}^{(1)}(x)$ |  | 1 | 1 | $\alpha$ | 0 | 1 | $1-\alpha$ | $\alpha$
Half-Normal: $2\,\mathcal{N}(x)$ |  | $\sqrt{2}$ | 1 | 2 | 0 | $\sqrt{2}$ | -1 | 2
Weibull: $\text{Wb}(x;k)$ |  | 1 | $k$ | $k$ | 0 | 1 | 0 | $k$
Exponential: $2\,L(x)$ |  $\text{Wb}(x;k=1)$ | 1 | 1 | 1 | 0 | 1 | 0 | 1
$\text{Rayleigh}(x)$ |  $\text{Wb}(\frac{x}{\sqrt{2}};k=2)$ | $\sqrt{2}$ | 2 | 2 | 0 | $\sqrt{2}$ | 0 | 2
Gamma: $\Gamma(\frac{x}{\sigma};s)$ |  | $\sigma$ | $s$ | 1 | 0 | $\sigma$ | $s-1$ | 1
$\chi_{k}(x)$ |  $\Gamma(\frac{x^{2}}{2};\frac{k}{2})$ | $\sqrt{2}$ | $k$ | 2 | 0 | $\sqrt{2}$ | $k-2$ | 2
$\chi_{k}^{2}(x)$ |  $\Gamma(\frac{x}{2};\frac{k}{2})$ | 2 | $\frac{k}{2}$ | 1 | 0 | 2 | $\frac{k}{2}-1$ | 1
$\text{GenGamma}(\frac{x}{\sigma};s,c)$ |  $\Gamma((\frac{x}{\sigma})^{c};s)$ | $\sigma$ | $sc$ | $c$ | 0 | $\sigma$ | $c(s-1)$ | $c$
or $f_{\text{\tiny GG}}(x;\sigma,d,p)$ |  | $\sigma$ | $d$ | $p$ | 0 | $\sigma$ | $d-p$ | $p$
${\text{IG}}(x;k)$ |  | 1 | $-k$ | $-1$ | 0 | 1 | $-k+1$ | $-1$
${\text{IWb}}(x;k)$ |  | 1 | $-k$ | $-k$ | 0 | 1 | 0 | $-k$
Table 1.  Mapping classic distributions to $\mathfrak{N}_{0}(x;\sigma,d,p)$.
$\mathcal{E}_{\alpha}^{(1)}(x)=\exp(-{x^{\alpha}})/\Gamma\left(\frac{1}{\alpha}+1\right)$
is a one-sided exponential power distribution.

###  5.4. Mapping between GSC and GG at alpha = 1/2

We prove the second line of (21) as following.

###### Proof.

From the property mentioned below (130) that
$M_{\frac{1}{2}}(z)=\frac{2}{z}W_{-\frac{1}{2},0}(-z)=\frac{1}{\sqrt{\pi}}e^{-z^{2}/4}$,
the exponential function can be re-interpreted as
$e^{-z}=\sqrt{\pi}\,M_{\frac{1}{2}}(2\sqrt{z})$.

By a change of variable $z={\left(\frac{x}{a}\right)}^{p}$, GG is mapped to
GSC at $\alpha=1/2$ as

$\displaystyle f_{\text{\tiny GG}}(x;a,d,p)$
$\displaystyle=\frac{\sqrt{\pi}\,p/a}{\Gamma(\frac{d}{p})}{\left(\frac{x}{a}\right)}^{d-1}\,M_{\frac{1}{2}}\left(2{\left(\frac{x}{a}\right)}^{p/2}\right)$
$\displaystyle=\frac{\sqrt{\pi}\,p/a}{\Gamma(\frac{d}{p})}{\left(\frac{x}{a}\right)}^{d-1}\,{\left(\frac{x}{a}\right)}^{-p/2}W_{-\frac{1}{2},0}\left(-2{\left(\frac{x}{a}\right)}^{p/2}\right)$
$\displaystyle=\frac{\sqrt{\pi}\,p/a}{\Gamma(\frac{d}{p})}{\left(\frac{x}{a}\right)}^{d-p/2-1}\,W_{-\frac{1}{2},0}\left(-{\left(\frac{2^{2/p}\,x}{a}\right)}^{p/2}\right)$
$\displaystyle=\mathfrak{N}_{\frac{1}{2}}\left(x;\sigma=\frac{a}{2^{2/p}},d=d-\frac{p}{2},p=\frac{p}{2}\right)$

Hence, we reach the second line of (21). ∎

If a classic distribution can be mapped to $\mathfrak{N}_{0}(x;...)$, it can
also be mapped to $\mathfrak{N}_{\frac{1}{2}}(x;...)$. That is how we map the
$\chi$ distribution from
${\chi}_{k}(s)=\mathfrak{N}_{0}(s;\sigma=\sqrt{2},d=k,p=2)$ to its fractional
counterpart
${\chi}_{k}(s)=\mathfrak{N}_{\frac{1}{2}}(s;\sigma=\frac{1}{\sqrt{2}},d=k-1,p=1)$
in (63) that eventually leads to (22).

The $\alpha=1/2$ mapping is illustrated in Table 1.

|  |  GSC: $\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
---|---|---
Distribution (PDF) | Classic Equiv. | $\alpha$ | $\sigma$ | $d$ | $p$
Weibull: $\text{Wb}(x;k)$ |  | $\frac{1}{2}$ | $2^{-2/k}$ | $\frac{k}{2}$ | $\frac{k}{2}$
Gamma: $\Gamma(\frac{x}{\sigma};s)$ |  | $\frac{1}{2}$ | $\sigma/4$ | $s-\frac{1}{2}$ | $\frac{1}{2}$
$\chi_{k}(x)$ |  $\Gamma(\frac{x^{2}}{2};\frac{k}{2})$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $k-1$ | 1
$\chi_{k}^{2}(x)$ |  $\Gamma(\frac{x}{2};\frac{k}{2})$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $(k-1)/2$ | $\frac{1}{2}$
$\text{GenGamma}(\frac{x}{\sigma};s,c)$ |  $\Gamma((\frac{x}{\sigma})^{c};s)$ | $\frac{1}{2}$ | $2^{-2/c}\sigma$ | $(s-\frac{1}{2})c$ | $c/2$
or $f_{\text{\tiny GG}}(x;\sigma,d,p)$ |  | $\frac{1}{2}$ | $2^{-2/p}\sigma$ | $d-\frac{p}{2}$ | $\frac{p}{2}$
${\text{IG}}(x;k)$ |  | $\frac{1}{2}$ | 4 | $\frac{1}{2}-k$ | $-\frac{1}{2}$
${\text{IWb}}(x;k)$ |  | $\frac{1}{2}$ | $2^{2/k}$ | $-k/2$ | $-k/2$
Table 2. Mapping of some classic distributions to
$\mathfrak{N}_{\frac{1}{2}}(x;\sigma,d,p)$

###  5.5. GSC Hankel Integral

The Hankel integral establishes another connection between GSC and GG. We
begin with the Hankel integral of $F_{\alpha}(z)$ in (129) where
$F_{\alpha}(z)=W_{-\alpha,0}(-z)=\frac{1}{2\pi
i}\int_{H}dt\,\exp{(t-zt^{\alpha})}$. Hence, for $\alpha d/p\neq 0$, the
Hankel integral of GSC is

$\displaystyle\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
$\displaystyle=\frac{1}{2\pi
i}\int_{H}dt\,C\,{\left(\frac{x}{\sigma}\right)}^{d-1}\,\exp{\left(t-{\left(\frac{x}{\sigma}\right)}^{p}t^{\alpha}\right)}$
$\displaystyle=\frac{1}{2\pi
i}\int_{H}dt\,e^{t}\,C\,{\left(\frac{x}{\sigma}\right)}^{d-1}\,\exp{\left(-{\left(\frac{x\,t^{\alpha/p}}{\sigma}\right)}^{p}\right)}$
(53) $\displaystyle=\,\Gamma\left(\frac{\alpha d}{p}\right)\,\frac{1}{2\pi
i}\int_{H}dt\,\left(\frac{e^{t}}{t^{\alpha d/p}}\right)\,f_{\text{\tiny
GG}}\left(x;a=\frac{\sigma}{t^{\alpha/p}},d=d,p=p\right)$

This shows that GSC can be obtained from the Hankel integral over its GG
counterpart.

###  5.6. GSC Moments

GSC’s moment formula plays a foundational role since FCM and GSaS’s moment
formulas are derived from it.

######  Lemma 5.1.

The $n$-th moment of GSC is

(54) $\displaystyle\mathbb{E}(X^{n}|\mathfrak{N}_{\alpha}(x;\sigma,d,p))$
$\displaystyle=\sigma^{n}\,\frac{\Gamma(\frac{d}{p}\alpha)}{\Gamma(\frac{d}{p})}\frac{\Gamma(\frac{n+d}{p})}{\Gamma(\frac{n+d}{p}\alpha)},$
$\displaystyle\,\,\text{for}\,\,\alpha\neq 0,d\neq 0.$

###### Proof.

The integral of the $n$-th moment is

$\displaystyle\mathbb{E}(X^{n}|\mathfrak{N}_{\alpha}(x;\sigma,d,p))$
$\displaystyle=\int_{0}^{\infty}dx\,x^{n}\,C\,{\left(\frac{x}{\sigma}\right)}^{d-1}\,{W}_{{-\alpha},0}\left(-{{\left(\frac{x}{\sigma}\right)}^{p}}\right)$
$\displaystyle=\sigma^{n+1}\,\frac{C}{p}\int_{0}^{\infty}dz\,\,{z}^{\frac{n+d}{p}-1}\,{W}_{{-\alpha},0}\left(-{z}\right),\quad\,\,\text{where}\,\,z={\left(\frac{x}{\sigma}\right)}^{p}$

This is basically the $\left(\frac{n+d}{p}\right)$-th moment of the Wright
function in (123). We arrive at

$\displaystyle\mathbb{E}(X^{n}|\mathfrak{N}_{\alpha}(x;\sigma,d,p))$
$\displaystyle=\sigma^{n+1}\,\frac{C}{p}\frac{\Gamma(\frac{n+d}{p})}{\Gamma(\frac{n+d}{p}\alpha)}$
$\displaystyle=\sigma^{n}\,\frac{\Gamma(\frac{d}{p}\alpha)}{\Gamma(\frac{d}{p})}\frac{\Gamma(\frac{n+d}{p})}{\Gamma(\frac{n+d}{p}\alpha)},$
$\displaystyle\,\,\text{for}\,\,\alpha\neq 0,d\neq 0.$

∎

Although the formula exists, whether a specific moment exists can be
complicated by the choice of $n$ relative to the shape parameters
$\alpha,d,p$.

We caution a common scenario in this paper that such formula needs special
handling due to $\Gamma(z)\approx 1/z$ when $z\to 0$. For instance, in this
case, $d=0$ or $n=-d$. This is documented in (137). There are numerous
situations like these, we may not mention them repeatedly in subsequent
encounters.

As a validation, when $\alpha\to 0$, it becomes the moments of $f_{\text{\tiny
GG}}(x;\sigma,d+p,p)$, as expected from Section 5.3:

$\displaystyle\mathbb{E}(X^{n}|\mathfrak{N}_{\alpha}(x;\sigma,d,p))$
$\displaystyle=\sigma^{n}\,\frac{\Gamma(\frac{n+d+p}{p})}{\Gamma(\frac{d+p}{p})}$

When $\alpha\to 1$, all gamma terms cancel out in $\mathbb{E}(X^{n})$, leaving
it to $\mathbb{E}(X^{n})=\sigma^{n}$. Therefore, the mean is $\sigma$, and the
variance is 0, which indicates it is a Dirac delta function
$\delta(x-\sigma)$.

###  5.7. Stable Count and Stable Vol Distributions

GSC traces its root to the discovery of the stable count distribution in
2017[16] and the stable vol distribution in 2020[17]. It is important to
reiterate the following relations.

The PDF of the original stable count distribution (SC) is defined as

(55)
$\mathfrak{N}_{\alpha}(\nu):=\frac{1}{\Gamma\left(\frac{1}{\alpha}+1\right)}\frac{1}{\nu}{L}_{\alpha}\left(\frac{1}{\nu}\right)=\frac{1}{\Gamma\left(\frac{1}{\alpha}+1\right)}{W}_{{-\alpha},0}\left(-{{\nu}^{\alpha}}\right)$

where ${L}_{\alpha}(x)$ is the PDF of the one-sided stable distribution.
$L_{\alpha}(x)=x^{-1}W_{-\alpha,0}(-x^{-\alpha})$. SC is subsumed to GSC as
$\mathfrak{N}_{\alpha}(\nu;\sigma=1,d=1,p=\alpha)$.

The stable vol distribution (SV) is a variant of SC where

(56)
$V_{\alpha}(s):=\frac{\sqrt{2\pi}\Gamma\left(\frac{2}{\alpha}+1\right)}{\Gamma\left(\frac{1}{\alpha}+1\right)}\mathfrak{N}_{\frac{\alpha}{2}}(2s^{2})=\frac{\sqrt{2\pi}}{\Gamma\left(\frac{1}{\alpha}+1\right)}{W}_{{-\frac{\alpha}{2}},0}\left(-{{(\sqrt{2}s)}^{\alpha}}\right).$

It is subsumed to GSC as
$\mathfrak{N}_{\frac{\alpha}{2}}(s;\sigma=1/\sqrt{2},d=1,p=\alpha)$.

The original idea of having two variants, SC and SV, came from the so-called
”Lambda decomposition”: transforming $e^{-{|z|}^{\alpha}}$ as a product
distribution with the Laplace distribution $L(x)$ and the normal distribution
$\mathcal{N}(x)$:

(59)
$\displaystyle\frac{1}{2\Gamma\left(\frac{1}{\alpha}+1\right)}e^{-{|z|}^{\alpha}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\int_{0}^{\infty}\frac{d\nu}{\nu}L\left(\frac{|z|}{\nu}\right)\,\mathfrak{N}_{\alpha}(\nu)&,\,\text{for}\,\,0<\alpha\leq
1.\\\
\displaystyle\int_{0}^{\infty}\frac{ds}{s}\mathcal{N}\left(\frac{|z|}{s}\right)\,V_{\alpha}(s)&,\,\text{for}\,\,0<\alpha\leq
2.\end{array}\right.$

It turns out the second line in (59) is very useful. First, the normal
distribution represents the Brownian noise. This is the early form of the
Gaussian mixture presented in Section 2.

Secondly, in the fractional $\chi$ notation,
$V_{\alpha}(x)=\chi_{\alpha,2}(x)$, which has two degrees of freedom. This
laid the groundwork for FCM.

Thirdly, it is in fact decomposing the CF of the $\alpha$-stable law into a
Gaussian mixture. It is used in the proof of Section 6 that follows.

In closing, several known fractional distributions are expressed in GSC in
Table 2.

|  |  GSC: $\mathfrak{N}_{\alpha}(x;\sigma,d,p)$
---|---|---
Distribution (PDF) | Wright Equiv. | $\alpha$ | $\sigma$ | $d$ | $p$
One-sided stable: $L_{\alpha}(x)$ |  $x^{-1}W_{-\alpha,0}(-x^{-\alpha})$ | $\alpha$ | 1 | 0 | $-\alpha$
SC: $\mathfrak{N}_{\alpha}(\nu)$ |  | $\alpha$ | 1 | 1 | $\alpha$
SV: $V_{\alpha}(s)$ |  | $\frac{\alpha}{2}$ | $\frac{1}{\sqrt{2}}$ | 1 | $\alpha$
M-Wright: $M_{\alpha}(z)$ |  $\frac{1}{\alpha z}W_{-\alpha,0}(-z)$ | $\alpha$ | 1 | 0 | 1
M-Wright: $\Gamma(\alpha)F_{\alpha}(z)$ |  $\Gamma(\alpha)W_{-\alpha,0}(-z)$ | $\alpha$ | 1 | 1 | 1
Table 3.  GSC mapping of several known fractional distributions in the
literature.

##  6\. The Skew-Gaussian Kernel and Fractional Chi-1

###  6.1. Derivation of the Skew-Gaussian Kernel

We prove how the kernel $g_{\alpha}^{\theta}(x,s)$ in (30) comes into
existence as following.

###### Proof.

The minus-log of $\alpha$-stable CF in Feller’s parameterization is defined as
(Section XVII.6, p.581-583 of [9]):

$\displaystyle\psi_{\alpha}^{\theta}(\zeta)$
$\displaystyle=\exp\left(\text{sgn}(\zeta)\frac{i\theta\pi}{2}\right){|\zeta|}^{\alpha}$
$\displaystyle=\left(\cos(\Theta)+i\sin(\Theta)\right){|\zeta|}^{\alpha},\quad\,\,\text{where}\,\,\Theta=\text{sgn}(\zeta)\frac{\theta\pi}{2}$
$\displaystyle\quad\text{and}\,\,0<\alpha\leq
2,|\theta|\leq\min\\{\alpha,2-\alpha\\}.$

Put it into the integral of the $\alpha$-stable PDF (See (F.35) of [18]), we
get

$\displaystyle L_{\alpha}^{\theta}({x})$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\zeta\,\exp(-\psi_{\alpha}^{\theta}(\zeta))\,e^{-ix\zeta}$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\zeta\,e^{-\cos(\Theta){|\zeta|}^{\alpha}}\,e^{-i(B+x\zeta)},\quad\,\,\text{where}\,\,B=\sin(\Theta){|\zeta|}^{\alpha}$

The imaginary part of the integral cancels out due to the sign of $\zeta$. The
real part of the integral can be simplified to the integral on the positive
axis $(\zeta\geq 0)$. We can remove $\text{sgn}(\zeta)$ from $\Theta$. Hence,

$\displaystyle L_{\alpha}^{\theta}({x})$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}d\zeta\,e^{-\cos(\Theta){\zeta}^{\alpha}}\,\cos(B+x\zeta),\quad\,\,\text{where}\,\,\Theta=\frac{\theta\pi}{2},B=\sin(\Theta){\zeta}^{\alpha}$

Next, we apply the Lambda decomposition (59) for the stable-vol
$V_{\alpha}(\zeta)$ on the $e^{-\cos(\Theta){\zeta}^{\alpha}}$ term. Let
$u=\cos(\Theta)^{1/\alpha}\zeta$, then
$e^{-\cos(\Theta){\zeta}^{\alpha}}=e^{-{u}^{\alpha}}$. We have

$\displaystyle L_{\alpha}^{\theta}({x})$
$\displaystyle=\int_{0}^{\infty}d\zeta\,\cos(B+x\zeta)\,\int_{0}^{\infty}\frac{ds}{s}\left(\frac{1}{\sqrt{2\pi}}e^{-(u/s)^{2}/2}\right)\left(\frac{2\Gamma(\frac{1}{\alpha}+1)}{\pi}V_{\alpha}(s)\right)$
(60) $\displaystyle=\int_{0}^{\infty}s\,ds\,\left(\frac{1}{\pi
s}\int_{0}^{\infty}d\zeta\,\cos(B+x\zeta)e^{-(u/s)^{2}/2}\right)\left(\frac{2\Gamma(\frac{1}{\alpha}+1)}{\sqrt{2\pi}}s^{-1}V_{\alpha}(s)\right)$

The terms inside the first big parenthesis of (60) is the skew-Gaussian kernel
in its raw format:

$\displaystyle g_{\alpha}^{\theta}(x,s)$ $\displaystyle=\frac{1}{\pi
s}\int_{0}^{\infty}d\zeta\,\cos(B+x\zeta)e^{-(u/s)^{2}/2},\quad\,\,\text{where}\,\,u=\cos(\Theta)^{1/\alpha}\zeta$

Make one more change of variable: $t=u/s=\cos(\Theta)^{1/\alpha}\zeta/s$, and
$B=\tan(\Theta){(st)}^{\alpha}$, we arrive at the final form presented in (30)
as

$\displaystyle g_{\alpha}^{\theta}(x,s)$
$\displaystyle=\frac{1}{q\pi}\int_{0}^{\infty}dt\,\cos\left(\tau\,{(st)}^{\alpha}+\frac{x}{q}\,st\right)e^{-t^{2}/2}\quad(s\geq
0)$
$\displaystyle\,\,\text{where}\,\,q=\cos(\theta\pi/2)^{1/\alpha},\,\,\tau=\tan(\theta\pi/2)$

∎

Next, we prove that $L_{\alpha}^{\theta}({x})$ is indeed
$\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\overline{\chi}_{\alpha,1}(s)$.

###### Proof.

The terms inside the second big parenthesis of (60) is simplified by (56),

(61)
$\displaystyle\left(\frac{2\Gamma(\frac{1}{\alpha}+1)}{\sqrt{2\pi}}s^{-1}V_{\alpha}(s)\right)$
$\displaystyle=2s^{-1}{W}_{{-\frac{\alpha}{2}},0}\left(-{{(\sqrt{2}s)}^{\alpha}}\right)$
$\displaystyle=\mathfrak{N}_{\frac{\alpha}{2}}(s;\sigma=\frac{1}{\sqrt{2}},d=0,p=\alpha)$
$\displaystyle=\overline{\chi}_{\alpha,1}(s)$

So according to (61) and (64), the $\alpha$-stable PDF (60) becomes a ratio
distribution of the kernel and an FCM of one degree of freedom:

(62) $\displaystyle L_{\alpha}^{\theta}({x})$
$\displaystyle=\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\overline{\chi}_{\alpha,1}(s)$

∎

###  6.2. Some Properties of the Skew Kernel

Some properties of $g_{\alpha}^{\theta}(x,s)$ are analyzed in this section.
This will help us better understand its behavior intuitively, as well as
provide a better implementation.

First, the integral inside $g_{\alpha}^{\theta}(x,s)$ is bounded by the
Gaussian integral $\int_{0}^{\infty}dt\,e^{-t^{2}/2}$, where $e^{-t^{2}/2}$ is
the upper bound to the absolute of the integrand since $|\cos(z)|\leq 1$.
Therefore, the contribution of the integrand decreases rapidly as $t$ gets
larger. This is the main reason such representation is chosen. Implementation-
wise, it is better to cut off the integral at a certain large $t$ (e.g. 100).

Indeed, when $\theta=0$, the Gaussian integral becomes exact, that is,
$g_{\alpha}^{0}(x,s)=\mathcal{N}(xs)$; and $L_{\alpha}^{0}(x)$ becomes
symmetric.

###### Proof.

When $\theta=0$, $q=1$ and $\tau=0$. Therefore,

$\displaystyle g_{\alpha}^{0}(x,s)$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}dt\,\cos\left(xst\right)e^{-t^{2}/2}$

From (142), this is $\frac{1}{\sqrt{2\pi}}e^{-(xs)^{2}/2}=\mathcal{N}(xs)$. ∎

When $\theta\neq 0$, some parts of $g_{\alpha}^{\theta}(x,s)$ can be negative.
Hence, it is not a distribution per se.

When $\alpha=1$, we have
$g_{1}^{\theta}(x,s)=\frac{1}{q}\,\mathcal{N}\left(\left(\tau+\frac{x}{q}\right)s\right)$,
and $L_{1}^{\theta}(x)$ becomes symmetric over
$x_{0}=-q\tau=-\sin(\theta\pi/2)$. That is, the effect of $\theta$ is no
longer skewness, but a shift in location.

###### Proof.

When $\alpha=1$, the argument of $\cos()$ in (30) becomes
$\tau\,{(st)}+\frac{x}{q}\,st=Yt$ where $Y=\left(\tau+\frac{x}{q}\right)s$.
Therefore, from (142),

$\displaystyle g_{1}^{\theta}(x,s)$
$\displaystyle=\frac{1}{q\pi}\int_{0}^{\infty}dt\,\cos\left(Yt\right)e^{-t^{2}/2}$
$\displaystyle=\frac{1}{q}\mathcal{N}(Y)=\frac{1}{q}\,\mathcal{N}\left(\left(\tau+\frac{x}{q}\right)s\right)$

∎

At $s=0$, $g_{\alpha}^{\theta}(x,0)$ is simply a constant $1/\sqrt{2\pi}q$.
This is straightforward since
$g_{\alpha}^{\theta}(x,0)=\frac{1}{q\pi}\int_{0}^{\infty}dt\,e^{-t^{2}/2}$.
Furthermore, for a given $(\alpha,\theta)$ pair,
$|g_{\alpha}^{\theta}(x,s)|\leq 1/\sqrt{2\pi}q$ for all $x$’s and $s$’s.

Next we prove:

######  Lemma 6.1.

For a given $s>0$, $\int_{-\infty}^{\infty}g_{\alpha}^{\theta}(x,s)\,dx=1/s$.

###### Proof.

We integrate the $\alpha$-stable PDF (62) by the full range of $x$, and it
should be one:

$\displaystyle\int_{-\infty}^{\infty}L_{\alpha}^{\theta}({x})\,dx$
$\displaystyle=\int_{-\infty}^{\infty}\,dx\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\overline{\chi}_{\alpha,1}(s)=1$

Rearranging the double integrals, it becomes

$\displaystyle\int_{0}^{\infty}s\,ds\,\left(\int_{-\infty}^{\infty}g_{\alpha}^{\theta}(x,s)\,dx\right)\,\overline{\chi}_{\alpha,1}(s)$
$\displaystyle=1$

Since $\overline{\chi}_{\alpha,1}(s)$ is a PDF, we have
$\int_{0}^{\infty}\overline{\chi}_{\alpha,1}(s)\,ds=1$. This forces the
integral inside the big parenthesis to be $1/s$, which is
$\int_{-\infty}^{\infty}g_{\alpha}^{\theta}(x,s)\,dx=1/s$.

∎

##  7\. FCM: Fractional Chi-Mean Distribution

FCM is the core of the innovation in this paper. We elaborate the thought
process that led to FCM in this section. There is a theoretical side of
thinking. There is also the aspect of design, mainly on how to choose the
scale of standardization such that it is consistent at both $k=1$ and
$k\to\infty$ limits.

###  7.1. Derivation of Fractional Chi

First of all, after many experiments (elaborated here and also the
subordination results in Appendix B), I’ve come to realization that we need to
use a ratio distribution (122), instead of a product distribution (121), to
describe the inner structures of both Student’s t and $\alpha$-stable. This is
a paradigm change.

Secondly, after many analyses, I’ve settled down on using a normal
distribution, instead of other alternatives, such as Laplace or Cauchy, as the
unit distribution in the middle. This has been formalized into the continuous
Gaussian mixture in Section 2.

The advantage of a normal distribution is obvious from the perspective of
stochastic calculus. Whatever distributional equation we come up with can be
translated to the random variable generation directly. Symbolically,
$L(x)=\int_{0}^{\infty}sds\mathcal{N}(sx)S(s)$ in the former becomes
$dL_{t}=dW_{t}/S_{t}$ in the later, where $dW_{t}$ is the Brownian noise.
Section 11 is dedicated to the subject of random variable generation, that is,
how $S_{t}$ can be generated.

Thirdly, for the expression of the normal distribution, we’ve evaluated two
options: (a) use the notion of variance (which we like to use $\nu$ for it in
(59)); or (b) use the notion of volatility (that is, squared root of variance,
which we like to use $s$ for it in (59)). After many thoughts, we didn’t chose
the variance route.

We wouldn’t say the last point is totally set in stone. Searches on ”chi-
squared distribution” vs ”chi distribution” showed that people like to talk
about sum of square of variables, instead of squared root of that sum. But we
find that our formula is better presented by the later, instead of the former.

During the following discussions, we assume $k>0$. $k$ has the ordinary
physical meaning as the _degrees of freedom_.

Now let’s begin with an expression of Student’s t distribution $t_{k}$ as the
ratio of a normal and the root-mean of a chi-squared:
$t_{k}\sim\mathcal{N}/\sqrt{\chi^{2}_{k}/k}$ as described in [28] and [31].

We prefer to combine the root-mean-square together. Therefore, we define the
$\chi$-mean distribution as
$\overline{\chi}_{k}=\sqrt{\chi^{2}_{k}/k}=\chi_{k}/\sqrt{k}$, which turns out
to be a GSC:

(63) $\displaystyle\overline{\chi}_{k}(s)$
$\displaystyle:=\mathfrak{N}_{\frac{1}{2}}\left(s;\sigma=\frac{1}{\sqrt{2k}},d=k-1,p=1\right)$

Next, we consider the more complicated matter of $\alpha$-stable. In Section
6, after a lengthy derivation, we arrive at (61), where we find ourselves
dealing with a similar GSC:

(64) $\displaystyle\overline{\chi}_{\alpha,1}(s)$
$\displaystyle:=\mathfrak{N}_{\frac{\alpha}{2}}(s;\sigma=\frac{1}{\sqrt{2}},d=0,p=\alpha)$

Here $\overline{\chi}_{\alpha,1}$ has a physical interpretation: It is the
$\alpha$-stable’s $\chi$ distribution with one degree of freedom. This
interpretation will be used in (66) below.

By comparing the GSC parameters between (63) and (64), it becomes obvious that
it is possible to merge $\overline{\chi}_{k}(s)$ and
$\overline{\chi}_{\alpha,1}(s)$ carefully and come up with a super
distribution family nicknamed FCM. The result is presented in (69). But it
takes quite some reasonings to get there…

###  7.2. Fractional Chi-1

In Lemma 3.3, we mentioned that $\overline{\chi}_{\alpha,1}$ is the fractional
version of classic $\chi_{1}$ and it is the building block of SaS since
$P_{\alpha}=L_{\alpha}^{0}\sim\mathcal{N}/\overline{\chi}_{\alpha,1}$, or in a
distributional form:

$\displaystyle P_{\alpha}(x):=L_{\alpha}^{0}(x)$
$\displaystyle=\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\overline{\chi}_{\alpha,1}(s)$

Compare it to $t_{k}\sim\mathcal{N}/\chi_{k}$. We can re-interpret
$P_{\alpha}$ as a sort of t-statistics in the fractional space. And
$\overline{\chi}_{\alpha,1}$ is the fractional _chi of one degree of freedom_
666 Wikipedia has a paragraph for this subject:
https://en.wikipedia.org/wiki/Proofs_related_to_chi-
squared_distribution#Derivation_of_the_pdf_for_one_degree_of_freedom.

The PDF of $\overline{\chi}_{\alpha,1}$ has some peculiar behaviors (listed
under Lemma 3.3) that span from a Dirac delta function, to a half-normal
distribution, to $1/x$ type. This affects the behavior of SaS directly. We
prove them here.

First, we show $\overline{\chi}_{2,1}(x)$ is a Dirac delta function. From
FCM’s moment formula (71), $\mathbb{E}(X^{n}|\overline{\chi}_{2,1})=2^{-n/2}$.
Its first moment is $\frac{1}{\sqrt{2}}$. Its second moment is $\frac{1}{2}$.
Its variance is zero. So it is a Dirac delta function at $\frac{1}{\sqrt{2}}$,
aka $\overline{\chi}_{2,1}(x)=\delta\left(x-1/\sqrt{2}\right)$.

Secondly, we show $\overline{\chi}_{1,1}(x)$ is a half-normal distribution.
From (69), its PDF is $2x^{-1}W_{-\frac{1}{2},0}(-\sqrt{2}x)$. From Section
5.4, it becomes $\sqrt{2/\pi}\,e^{-x^{2}/2}$.

Thirdly, we prove that $\overline{\chi}_{\alpha,1}(x)\approx\alpha/(ex)$, and
subsequently,
$P_{\alpha}(x)\approx\overline{\chi}_{\alpha,1}(x)/2\approx\alpha/(2ex)$, when
$\alpha$ is very small but $x$ is not too small.

###### Proof.

We use the result of GSC at $\alpha\to 0$ in Section 5.3. Combine (64) with
(52), we obtain

$\displaystyle\overline{\chi}_{\alpha,1}(x)$ $\displaystyle\to f_{\text{\tiny
GG}}(x;\sigma=\frac{1}{\sqrt{2}},d=\alpha,p=\alpha)$
$\displaystyle=\frac{\alpha}{x}{(\sqrt{2}x)}^{\alpha}\exp\left(-{(\sqrt{2}x)}^{\alpha}\right)$

The condition that $\alpha$ is very small but $x$ is not too small means that
${(\sqrt{2}x)}^{\alpha}\approx 1$. That is, $\alpha|\log(x)|\to 0$. Under such
condition, $\overline{\chi}_{\alpha,1}(x)\approx\alpha/(ex)$.

For $P_{\alpha}(x)$ at $\alpha\to 0$, we have

$\displaystyle P_{\alpha}(x)$
$\displaystyle=\int_{0}^{\infty}sds\,\mathcal{N}(sx)\,\bar{\chi}_{\alpha,1}(s)$
$\displaystyle=(\alpha/e)\,\int_{0}^{\infty}ds\,\mathcal{N}(sx)$
$\displaystyle=\alpha/(2ex)$

∎

This example illustrates it is increasingly difficult to calculate
$P_{\alpha}(x)$ through integrating $s\,\overline{\chi}_{\alpha,1}(s)$
numerically when $\alpha$ is small (e.g. $\alpha<0.3$) and $x\to 0$.

###  7.3. Physical Interpretation of Fractional Chi

To gain physical understanding on $\overline{\chi}_{\alpha,k}$, let’s repeat
how the $\chi^{2}$ distribution is derived classically, as described in [27],
but adapt it to the fractional interpretation needed for our purpose.

A fractionally generalized chi-distributed random variable $R$ with $k$
degrees of freedom is defined as the squared root of the sum of $k$ squared
random variables of the $S_{\alpha}$ distribution, which shall be clarified
later that $S_{\alpha}$ is $\overline{\chi}_{\alpha,1}$. So we have

$\displaystyle X_{1},...,X_{k}\sim S_{\alpha}$ $\displaystyle\implies
R:=\sqrt{\sum_{i=1}^{k}X_{i}^{2}}$

Let $x_{1},...,x_{k}$ be the values of $X_{1},...,X_{k}$ and consider
$(x_{1},...,x_{k})$ to be a point in the $k$-dimensional space. Define the
distance to that point:

$\displaystyle r=\sqrt{\sum_{i=1}^{k}x_{i}^{2}}$

Let $f_{R}(r)$ and $F_{R}(r)$ be the PDF and CDF of $R$. Their relation is

$\displaystyle F_{R}(r+dr)-F_{R}(r)$ $\displaystyle=f_{R}(r)dr$
$\displaystyle=\int_{V}\prod_{i=1}^{k}S_{\alpha}(x_{i})dx_{i}$

where $S_{\alpha}(x_{i})$ is the PDF of the $S_{\alpha}$ distribution, and $V$
is the volume on the $(k-1)$-dimensional surface in the $k$-space at the
radius $r$.

Assume we can refactor the above integral such that

$\displaystyle f_{R}(r)dr$
$\displaystyle=\left(\int_{V}dx_{1}...dx_{k}\right)\left(\prod_{i=1}^{k}S_{\alpha}(x_{i})\right)$

where we deal with the two big parentheses separately.

The first part is an integral of the surface area of the $(k-1)$-dimensional
sphere with radius $r$, which is

(65) $\displaystyle A(r)dr$
$\displaystyle=\left(\int_{V}dx_{1}...dx_{k}\right)=2\,r^{k-1}\frac{\pi^{k/2}}{\Gamma(k/2)}\,dr$

For the second part, we argue it should just be $\overline{\chi}_{\alpha,1}$
in (64):

(66) $\displaystyle\left(\prod_{i=1}^{k}S_{\alpha}(x_{i})\right)$
$\displaystyle\sim\overline{\chi}_{\alpha,1}(r)=\frac{2}{r}\,W_{\frac{\alpha}{2},0}\left(-{(\sqrt{2}r)}^{\alpha}\right)$

Combining (65) and (66), we arrive at the PDF for $R$ (apart from a
normalization constant):

$\displaystyle f_{R}(r)$ $\displaystyle\propto
r^{k-2}\,W_{\frac{\alpha}{2},0}\left(-{(\sqrt{2}r)}^{\alpha}\right)$
$\displaystyle=\mathfrak{N}_{\frac{\alpha}{2}}(r;\sigma=\frac{1}{\sqrt{2}},d=k-1,p=\alpha)$

Therefore, $f_{R}(r)$ is the fractional-chi $\chi_{\alpha,k}(r)$ that we seek.

Note that we glossed over the constant term. We picked up the $r^{k-1}$ factor
from $A(r)$ but ignored the rest of it. This ”surface area of the
$(k-1)$-dimensional sphere” is more complicated than a simple Euclidean
sphere. This topic is left for future research.

###  7.4. From Fractional-Chi to Fractional Chi-Mean

Now we have
$\chi_{\alpha,k}(x)=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma=\frac{1}{\sqrt{2}},d=k-1,p=\alpha)$.
We need to ”standardize” its mean, so to speak. Our cue is that
$\overline{\chi}_{k}=\chi_{k}/\sqrt{k}$ works for Student’s t. A simple
multiplier of $1/\sqrt{k}$ for $\sigma$ suffices. The question is: What works
for $\chi_{\alpha,k}$ in general?

We examine the mean of $\chi_{\alpha,k}$. From (54), its mean is

$\displaystyle\mathbb{E}(X|\chi_{\alpha,k})$
$\displaystyle=\frac{1}{\sqrt{2}}\,\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{k}{\alpha})}{\Gamma(\frac{k}{2})}$

At $k=1$,
$\mathbb{E}(X|\chi_{\alpha,1})=\sqrt{\frac{2}{\pi}}\,\Gamma\left(\frac{1}{\alpha}+1\right)$,
which we want to preserve for $\alpha$-stable.

Its large $k$ asymptotic is

(67) $\displaystyle\mathbb{E}(X|\chi_{\alpha,k})$ $\displaystyle\sim
k^{1/\alpha-1/2}\,\alpha^{-1/\alpha},\quad k\gg 1;$

whose $k$ dependency we would like to eliminate so that the mean doesn’t go to
0 or infinity.

So our requirement for FCM is to modify the $\sigma$ parameter in
$\chi_{\alpha,k}$, according to the $k$ component of $\chi_{\alpha,k}$’s
asymptotic mean, such that FCM has an asymptotic mean independent of $k$.

The solution is the following. Define

(68) $\displaystyle\sigma_{\alpha,k}$
$\displaystyle:=\frac{k^{1/2-1/\alpha}}{\sqrt{2}}$

which has the desired properties: $\sigma_{1,k}=1/\sqrt{2k}$, and
$\sigma_{\alpha,1}=1/\sqrt{2}$, in order to preserve both Student’s t and
$\alpha$-stable distributions.

Hence, FCM is constructed as

(69) $\displaystyle\overline{\chi}_{\alpha,k}(x)$
$\displaystyle:=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma=\sigma_{\alpha,k},d=k-1,p=\alpha)$
$\displaystyle=\mathfrak{N}_{\frac{\alpha}{2}}(x;\sigma=\frac{1}{\sqrt{2}}\left(k^{1/2-1/\alpha}\right),d=k-1,p=\alpha)$
$\displaystyle=\frac{\alpha\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\,{\left(\frac{\sqrt{2}}{k^{1/2-1/\alpha}}\right)}^{k-1}{x}^{k-2}\,W_{\frac{\alpha}{2},0}\left(-{\left(\frac{\sqrt{2}\,x}{k^{1/2-1/\alpha}}\right)}^{\alpha}\right)$

To verify, its large $k$ asymptotic mean doesn’t have $k$ dependency: (See
Section 7.8)

$\displaystyle\lim_{k\to\infty}\mathbb{E}(X|\overline{\chi}_{\alpha,k})$
$\displaystyle=\alpha^{-1/\alpha}$

This meets our design goal.
$\mathbb{E}(X|\overline{\chi}_{\alpha,1})=\mathbb{E}(X|\chi_{\alpha,1})$ and
$\lim_{k\to\infty}\mathbb{E}(X|\overline{\chi}_{1,k})=1$ as expected. And it
is nice to have $\mathbb{E}(X|\overline{\chi}_{2,k})=1/\sqrt{2}$ for every
$k$.

Interpretation: The $\alpha$-stable law states that

$\displaystyle X_{1}+X_{2}+...+X_{k}$ $\displaystyle\sim k^{1/\alpha}X$

Therefore, the $\left(k^{1/2-1/\alpha}\right)$ term can be viewed as the
relative scale (ratio) between the normal scaling factor $k^{1/2}$ and
$\alpha$-stable scaling factor $k^{1/\alpha}$. That is what we added to
$\sigma_{\alpha,k}$ as the way to ”demean” the $k$ dependency.

###  7.5. FCM Hankel Integral

From (53), make the following substitutions: $\alpha/p=1/2,d=k-1,\alpha
d/p=(k-1)/2,p=\alpha$, and $\sigma=\sigma_{\alpha,k}$. The Hankel integral of
FCM for $k>0$ is

(70) $\displaystyle\overline{\chi}_{\alpha,k}({x})$
$\displaystyle=\,\Gamma\left(\frac{k-1}{2}\right)\,\frac{1}{2\pi
i}\int_{H}dt\,\left(\frac{e^{t}}{t^{(k-1)/2}}\right)\,f_{\text{\tiny
GG}}\left(x;a=\frac{\sigma_{\alpha,k}}{\sqrt{t}},d=k-1,p=\alpha\right)$

This expression requires $k\neq 1$.

###  7.6. FCM Moments

FCM’s moment is a special case of GSC’s moment, which is carried out in
Section 5.6.

For $k>0$, based on (22) and (54), substitute $\alpha/p=1/2,d=k-1,p=\alpha$
and $\sigma=\sigma_{\alpha,k}$, and we obtain FCM’s moments as

(71) $\displaystyle\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,k})$
$\displaystyle={\left(\frac{k^{1/2-1/\alpha}}{\sqrt{2}}\right)}^{n}\,\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{n+k-1}{\alpha})}{\Gamma(\frac{n+k-1}{2})},$
$\displaystyle\,\,\text{for}\,\,\alpha\neq 0,k>0,k\neq 1.$

When $k=1$ or $k+n=1$, it requires typical handling of gamma functions at zero
(137).

For $k<0$, based on (23) and (54), substitute $\alpha/p=-1/2,d=k,p=-\alpha$
and $\sigma=1/\sigma_{\alpha,k}$, and we obtain FCM’s moments as

(72) $\displaystyle\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,k})$
$\displaystyle={\left(\frac{{|k|}^{1/2-1/\alpha}}{\sqrt{2}}\right)}^{-n}\,\frac{\Gamma(\frac{|k|}{2})}{\Gamma(\frac{|k|}{\alpha})}\frac{\Gamma(\frac{|k|-n}{\alpha})}{\Gamma(\frac{|k|-n}{2})},$
$\displaystyle\,\,\text{for}\,\,\alpha\neq 0,k<0.$

When $n=|k|$, it requires typical handling of gamma functions at zero (137).
Higher moments may not exist when $n>|k|$.

As a validation, when $\alpha=1$, (71) is reduced to the moment formula of a
$\chi$ distribution with $k$ degrees of freedom and scale of $1/\sqrt{k}$ due
to (136).

FCM’s first moment is directly related to GSaS peak PDF in (76). Numeric
calculation of FCM’s moments is important in that it directly impacts the
accuracy of GAS PDF calculation, especially at small $\alpha$ and when $x$ is
near zero.

###  7.7. FCM Reflection Formula

We prove the reflection formula shown in (25) and (26).

$\displaystyle\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,-k})$
$\displaystyle=\frac{\mathbb{E}(X^{-n+1}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})},\,\,k>0.$

The approach is to show that the moments evaluated from the RHS via (71) is
the same as those of the LHS from (72) with $k$ replaced with $-k$.

###### Proof.

Assume $k>0$, the $n$-th moment on the LHS is

$\displaystyle\mathbb{E}(X^{n}|\text{LHS})$
$\displaystyle=\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,-k})={\left(\frac{{k}^{1/2-1/\alpha}}{\sqrt{2}}\right)}^{-n}\,\frac{\Gamma(\frac{k}{2})}{\Gamma(\frac{k}{\alpha})}\frac{\Gamma(\frac{k-n}{\alpha})}{\Gamma(\frac{k-n}{2})}$

The $n$-th moment on the RHS is

$\displaystyle\mathbb{E}(X^{n}|\text{RHS})$
$\displaystyle=\int_{0}^{\infty}\frac{x^{n-3}}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\,\overline{\chi}_{\alpha,k}\left(\frac{1}{x}\right)\,dx$

Make a change of variable $t=1/x$, we get

$\displaystyle\mathbb{E}(X^{n}|\text{RHS})$
$\displaystyle=\int_{0}^{\infty}\frac{t^{-n+1}}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\,\overline{\chi}_{\alpha,k}\left(t\right)\,dt$
$\displaystyle=\frac{\mathbb{E}(X^{-n+1}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}$

Hence, (26) is true if $\mathbb{E}(X^{n}|\text{RHS})$ =
$\mathbb{E}(X^{n}|\text{LHS})$.

Plug (71) to both the numerator and denominator,

$\displaystyle\mathbb{E}(X^{n}|\text{RHS})$
$\displaystyle=\left({\left(\frac{{|k|}^{1/2-1/\alpha}}{\sqrt{2}}\right)}^{-n+1}\,\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{-n+k}{\alpha})}{\Gamma(\frac{-n+k}{2})}\right)\bigg{/}\left({\left(\frac{{|k|}^{1/2-1/\alpha}}{\sqrt{2}}\right)}\,\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{k}{\alpha})}{\Gamma(\frac{k}{2})}\right)$
$\displaystyle={\left(\frac{{|k|}^{1/2-1/\alpha}}{\sqrt{2}}\right)}^{-n}\,\frac{\Gamma(\frac{k}{2})}{\Gamma(\frac{k}{\alpha})}\frac{\Gamma(\frac{k-n}{\alpha})}{\Gamma(\frac{k-n}{2})}$

It is clear that $\mathbb{E}(X^{n}|\text{RHS})$ and
$\mathbb{E}(X^{n}|\text{LHS})$ are the same.

∎

###  7.8. FCM at Infinite Degrees of Freedom

First, we establish the following lemma:

######  Lemma 7.1.

We use (138) to simplify (71) and obtain the asymptotic value of the $n$-th
moment for $k\gg 1$:

(73) $\displaystyle\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,k})$
$\displaystyle\sim{\left(\frac{{|k|}^{1/2-1/\alpha}}{\sqrt{2}}\right)}^{n}\,{\left(\frac{k}{2}\right)}^{-n/2}{\left(\frac{k}{\alpha}\right)}^{n/\alpha}={\alpha}^{-n/\alpha}$

It follows from the reflection rule that
$\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,-k})\sim{\alpha}^{n/\alpha}$. Notice
that the asymptotic value has no dependency on $k$.

The proof for Lemma 3.2 becomes very easy, which states that FCM becomes a
delta function at $x=m_{\alpha}$ when $k\gg 1$, where $m_{\alpha}$ is its
first moment. We prove it by showing $m_{\alpha}$ is ${\alpha}^{-1/\alpha}$
and its variance is zero.

###### Proof.

Let $m_{\alpha}={\alpha}^{-1/\alpha}$, we have
$\mathbb{E}(X^{n}|\overline{\chi}_{\alpha,k})\sim{(m_{\alpha})}^{n}$. Hence,
the first moment is obviously

$\displaystyle\mathbb{E}(X|\overline{\chi}_{\alpha,k})\sim{\alpha}^{-1/\alpha}$

The second moment is $\mathbb{E}(X^{2}|\overline{\chi}_{\alpha,k})\sim
m_{\alpha}^{2}$. Therefore, its variance is
$\mathbb{E}(X^{2}|\overline{\chi}_{\alpha,k})-m_{\alpha}^{2}=0$. FCM becomes a
delta function as $k\to\infty$.

On the negative side, the reflection rule ensures the same proof applies such
that
$\mathbb{E}(X^{2}|\overline{\chi}_{\alpha,-k})-\mathbb{E}(X|\overline{\chi}_{\alpha,-k})^{2}=0$.
FCM becomes a delta function as $-k\to-\infty$.

∎

##  8\. GSaS: Generalized Symmetric alpha-Stable Distribution

The PDF of a GSaS takes the form of an elegant ratio distribution:

$\displaystyle L_{\alpha,k}({x})$
$\displaystyle:=\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\overline{\chi}_{\alpha,k}(s)$

This two-sided distribution is one of the most important outcomes of this
work. We prove some closed form results in this section.

###  8.1. GSaS Peak PDF

The peak PDF of a GSaS at $x=0$ comes from the first moment of FCM in Section
7.6:

$\displaystyle L_{\alpha,k}({0})$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}s\,ds\,\overline{\chi}_{\alpha,k}(s)=\frac{1}{\sqrt{2\pi}}\mathbb{E}(X|\overline{\chi}_{\alpha,k})$
(76)
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\frac{k^{1/2-1/\alpha}}{2\sqrt{\pi}}\,\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{k}{\alpha})}{\Gamma(\frac{k}{2})},&\,\,k>0.\\\\[14.45377pt]
\displaystyle\frac{1}{\sqrt{\pi}{\left({|k|}^{1/2-1/\alpha}\right)}}\,\frac{\Gamma(\frac{|k|}{2})}{\Gamma(\frac{|k|}{\alpha})}\frac{\Gamma(\frac{|k|-1}{\alpha})}{\Gamma(\frac{|k|-1}{2})},&\,\,k<0.\end{array}\right.$

When $k={\pm}1$, it requires the typical special handling of the gamma
function at zero.

The standardized peak PDF is the peak PDF $L_{\alpha,k}({0})$ multiplied by
the standard deviation, $\sqrt{\mathbb{E}(X^{2}|L_{\alpha,k})}$, if it exists.
We argue that the standardized peak PDF is an important statistic since it is
invariant to the scale of the sample data. In finance, this means that the
standardized peak PDF measures the similarity of two data sets taken from two
periods of different volatility environments, or from two different assets
altogether.

Aligning the peak PDF was often impossible in the past due to the limited one
dimensional parameter space in either $\alpha$ or $k$. GSaS overcomes it with
a two-dimensional parameter space. This opens up much greater flexibility,
e.g., to match the standard deviation, the peak density, and the kurtosis
simultaneously.

The first validation of $L_{\alpha,k}({0})$ is that it becomes the peak PDF of
the original $\alpha$-stable distribution when $k\to 1$:

$\displaystyle L_{\alpha,1}(0)=P_{\alpha}(0)$
$\displaystyle=\frac{1}{\pi}\,\Gamma\left(1+1/\alpha\right)$

The second validation is that it becomes the peak PDF of the exponential power
distribution when $k\to-1$:

$\displaystyle L_{\alpha,-1}(0)=\mathcal{E}_{\alpha}(0)$
$\displaystyle=\frac{1}{2\Gamma\left(1+1/\alpha\right)}$

###  8.2. GSaS Moments

GSaS has closed form solution for all the moments. This is a great improvement
over the original $\alpha$-stable distribution.

######  Lemma 8.1.

The $n$-th moment of GSaS is

(77) $\displaystyle\mathbb{E}(X^{n}|L_{\alpha,k})$
$\displaystyle=\frac{2^{n/2}}{\sqrt{\pi}}\,\Gamma\left(\frac{n+1}{2}\right)\,\mathbb{E}(X^{-n}|\overline{\chi}_{\alpha,k}),\quad
k\neq 0$

when $n$ is even. The odd moments are zero.

Its closed form is

(80) $\displaystyle\mathbb{E}(X^{n}|L_{\alpha,k})$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\frac{2^{n}}{\sqrt{\pi}\left(k^{1/2-1/\alpha}\right)^{n}}\,\Gamma\left(\frac{n+1}{2}\right)\,\left(\frac{\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\frac{\Gamma(\frac{k-n-1}{\alpha})}{\Gamma(\frac{k-n-1}{2})}\right),&\,\,k>0.\\\\[14.45377pt]
\displaystyle\frac{{({|k|}^{1/2-1/\alpha})}^{n}}{\sqrt{\pi}}\,\Gamma\left(\frac{n+1}{2}\right)\,\left(\frac{\Gamma(\frac{|k|}{2})}{\Gamma(\frac{|k|}{\alpha})}\frac{\Gamma(\frac{|k|+n}{\alpha})}{\Gamma(\frac{|k|+n}{2})}\right),&\,\,k<0.\end{array}\right.$

When $k={\pm}1,n+1,-n$, it requires the typical special handling of the gamma
function at zero.

This formula is continuous and well-behaves when $k$ is _sufficiently large_ ,
but it gets very complicated when $k$ is small. This is a complexity inherent
from the $\alpha$-stable distribution.

###### Proof.

The integral of the $n$-th moment is

$\displaystyle\mathbb{E}(X^{n}|L_{\alpha,k})$
$\displaystyle=\int_{-\infty}^{\infty}dx\,x^{n}\,\int_{0}^{\infty}s\,ds\,\mathcal{N}(xs)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle=\int_{0}^{\infty}s\,ds\,\left(\int_{-\infty}^{\infty}dx\,x^{n}\,\mathcal{N}(xs)\right)\,\overline{\chi}_{\alpha,k}(s)$

Inside the big parenthesis is the $n$-th moment of a normal distribution (see
(144)), which is
$\frac{2^{n/2}}{\sqrt{\pi}}s^{-n-1}\Gamma\left(\frac{n+1}{2}\right)$.
Therefore, we get

$\displaystyle\mathbb{E}(X^{n}|L_{\alpha,k})$
$\displaystyle=\frac{2^{n/2}}{\sqrt{\pi}}\,\Gamma\left(\frac{n+1}{2}\right)\,\int_{0}^{\infty}s^{-n}\,ds\,\overline{\chi}_{\alpha,k}({s})$

The remaining integral is the ”$-n$”-th moment of FCM. Hence, we get (77).

To obtain the first line of (80), copy (71) into (77), and replace every $n$
with $-n$.

Similarly, copy (72) into (77), replace every $n$ with $-n$, and we reach the
second line of (80).

∎

GSaS Variance: The second moment $m_{2}$, aka the variance, is

$\displaystyle m_{2}=\mathbb{E}(X^{2}|L_{\alpha,k})$
$\displaystyle=\mathbb{E}(X^{-2}|\overline{\chi}_{\alpha,k})$

In general, $m_{2}\neq 1$ for finite $k$.

At $k\gg 1$, according to (73), the variance of the normal-like distribution
is

$\displaystyle\lim_{k\to\infty}m_{2}$ $\displaystyle={\alpha}^{2/\alpha}$
$\displaystyle\lim_{k\to-\infty}m_{2}$ $\displaystyle={\alpha}^{-2/\alpha}$

Both are independent of $k$. This is the intended result from Section 7.8.

###  8.3. GSaS Kurtosis

Let $m_{4}=\mathbb{E}(X^{4}|L_{\alpha,k})$, it is straightforward to get the
kurtosis as

(81) $\displaystyle\frac{m_{4}}{m_{2}^{2}}$
$\displaystyle=3\,\frac{\mathbb{E}(X^{-4}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X^{-2}|\overline{\chi}_{\alpha,k})^{2}}$

The excess kurtosis is $\text{exKurt}=m_{4}/m_{2}^{2}-3$.

For $k>0$, the closed form is

$\displaystyle\frac{m_{4}}{m_{2}^{2}}$ $\displaystyle=\displaystyle
3\,{\left(\frac{\Gamma\left(\frac{k-1}{2}\right)}{\Gamma\left(\frac{k-1}{\alpha}\right)}\frac{\Gamma\left(\frac{k-5}{\alpha}\right)}{\Gamma\left(\frac{k-5}{2}\right)}\right)}\bigg{/}{{\left(\frac{\Gamma\left(\frac{k-1}{2}\right)}{\Gamma\left(\frac{k-1}{\alpha}\right)}\frac{\Gamma\left(\frac{k-3}{\alpha}\right)}{\Gamma\left(\frac{k-3}{2}\right)}\right)}^{2}}$
(82)
$\displaystyle=3\,\left(\frac{k-5}{k-3}\right)\frac{\Gamma(s(k-5))\Gamma(s(k-1))}{\Gamma(s(k-3))^{2}},\quad\,\,\text{where}\,\,\,s=1/\alpha.$

We prove the linear relation of excess kurtosis in (37) for $k\gg 1$ as
follows.

###### Proof.

In Figure 5, notice the singular point at $s=1/2,k=3$. Hence we make a change
of variable by $k_{3}:=k-3$:

$\displaystyle\frac{m_{4}}{m_{2}^{2}}$
$\displaystyle=3\,\left(1-\frac{2}{k_{3}}\right)\frac{\Gamma(s(k_{3}-2))\Gamma(s(k_{3}+2))}{\Gamma(sk_{3})^{2}}$

Next, the gamma functions are expanded via the Sterling’s formula (139) for
large $k_{3}$. It is simplified to

$\displaystyle\frac{m_{4}}{m_{2}^{2}}$
$\displaystyle=3\,{\left(1-\frac{2}{k_{3}}\right)}^{sk_{3}-2s+1/2}\,{\left(1+\frac{2}{k_{3}}\right)}^{sk_{3}+2s-1/2}$

Take the log on both sides, and retain the Taylor series of $\log(1+x)$ up to
$k_{3}^{-2}$. We have

$\displaystyle\log{\left(\frac{m_{4}}{3\,m_{2}^{2}}\right)}$
$\displaystyle=\frac{4}{k_{3}}{\left(s-\frac{1}{2}\right)}$

Since $m_{4}/m_{2}^{2}=3+\text{exKurt}$ where exKurt is the excess kurtosis,
we arrive at the proof that

$\displaystyle{\left(s-\frac{1}{2}\right)}$
$\displaystyle=\left(\frac{k-3}{4}\right)\log{\left(1+\frac{\text{exKurt}}{3}\right)}$

∎

As $k\to\infty$, the excess kurtosis approaches zero. GSaS becomes a normal
distribution asymptotically, $\mathcal{N}(0,m_{2})$. Only when $\alpha=1$,
$\lim_{k\to\pm\infty}m_{2}=1$. It becomes a $\mathcal{N}(0,1)$.

###  8.4. GSaS CDF

The CDF formula of GSaS is straightforward since
$\int_{-\infty}^{x}\,\mathcal{N}\left(z\right)dz$
$=\frac{1}{2}+\frac{1}{2}\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)$.

######  Lemma 8.2.

The CDF of a GSaS is

(83) $\displaystyle\Phi[L_{\alpha,k}](x)$
$\displaystyle:=\int_{-\infty}^{x}\,L_{\alpha,k}\left(z\right)dz=\frac{1}{2}+\frac{1}{2}\int_{0}^{\infty}\,ds\,\operatorname{erf}\left(\frac{xs}{\sqrt{2}}\right)\,\overline{\chi}_{\alpha,k}(s)$

###### Proof.

We begin with the double integral via the PDF, such that

$\displaystyle\Phi[L_{\alpha,k}](x)$
$\displaystyle=\int_{-\infty}^{x}\,L_{\alpha,k}\left(z\right)dz$
$\displaystyle=\int_{-\infty}^{x}\,dz\int_{0}^{\infty}s\,ds\,\mathcal{N}(zs)\,\overline{\chi}_{\alpha,k}(s)$

The order of the two integrals is switched, and we make a change of variable
$u=zs$:

$\displaystyle\Phi[L_{\alpha,k}](x)$
$\displaystyle=\int_{0}^{\infty}s\,ds\,\left(\int_{-\infty}^{x}\,dz\mathcal{N}(zs)\right)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle=\int_{0}^{\infty}\,ds\,\left(\int_{-\infty}^{xs}\,du\mathcal{N}(u)\right)\,\overline{\chi}_{\alpha,k}(s)$

The inner integral is replaced with the error function:

$\displaystyle\Phi[L_{\alpha,k}](x)$
$\displaystyle=\int_{0}^{\infty}\,ds\,\left(\frac{1}{2}+\frac{1}{2}\operatorname{erf}\left(\frac{xs}{\sqrt{2}}\right)\right)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle=\frac{1}{2}+\frac{1}{2}\int_{0}^{\infty}\,ds\,\operatorname{erf}\left(\frac{xs}{\sqrt{2}}\right)\,\overline{\chi}_{\alpha,k}(s)$

We arrive at the desired formula for the CDF. ∎

###  8.5. Fractional Hypergeometric Function

Due to the connection between the CDF of Student’s t and the Gauss
hypergeometric function $\,{}_{2}F_{1}\left(a,b;c;z\right)$ (defined in 15.2.1
of DLMF[6]), we propose a framework to extend $\,{}_{2}F_{1}$ fractionally.

######  Definition 8.1 (Fractional hypergeometric function (FHF)).

Let $M(b,c;z)$ be the confluent hypergeometric function, or called the Kummer
function, defined in 13.2.2 of DLMF. (Note we avoid using $a$ because it is
confusing to $\alpha$.)

The fractional hypergeometric function is defined in a ratio-distribution
style as

(84) $\displaystyle M_{\alpha,k}(b,c;x)$
$\displaystyle:=\sqrt{\frac{k}{2\pi}}\int_{0}^{\infty}s\,ds\,M\left(b,c;\frac{xks^{2}}{2}\right)\,\overline{\chi}_{\alpha,k}(s)$

Note that $x$ usually takes a negative value. This is due to the asymptotic
behavior: $M(b,c;z)\sim e^{z}z^{b-c}/\Gamma(b)$ as $z\to\infty$ (13.2.23 of
DLMF).

######  Lemma 8.3.

The CDF of a GSaS can be expressed by the fractional hypergeometric function
as

(85) $\displaystyle\Phi[L_{\alpha,k}](x)$
$\displaystyle=\frac{1}{2}+\frac{x}{\sqrt{k}}M_{\alpha,k}\left(\frac{1}{2},\frac{3}{2};-\frac{x^{2}}{k}\right)$

###### Proof.

The error function can be expressed by the Kummer function as (13.6.7 of DLMF)

$\displaystyle\operatorname{erf}(x)$
$\displaystyle=\frac{2x}{\sqrt{\pi}}\,M\left(\frac{1}{2},\frac{3}{2};-x^{2}\right)$

Substitute it to (83), we obtain (85). ∎

######  Lemma 8.4.

The fractional hypergeometric function subsumes the Gauss hypergeometric
function at $\alpha=1$:

(86) $\displaystyle M_{1,k}(b,c;x)\,B\left(\frac{k}{2},\frac{1}{2}\right)$
$\displaystyle=\,_{2}F_{1}\left(b,\frac{k+1}{2};c;x\right)$

where $B(a,b)$ is the beta function (8.17.3 of DLMF). Notice that $k$ plays
the role of the second parameter in $\,{}_{2}F_{1}$.

###### Proof.

Since
$\overline{\chi}_{1,k}(s)=\frac{k^{k/2}}{2^{k/2-1}\Gamma(k/2)}s^{k-1}e^{-ks^{2}/2}$,
(86) becomes

$\displaystyle\int_{0}^{\infty}ds\,M\left(b,c;\frac{xks^{2}}{2}\right)\,s^{k}\,e^{-ks^{2}/2}$
$\displaystyle=\frac{1}{2}\Gamma\left(\frac{k+1}{2}\right)\,\left(\frac{2}{k}\right)^{(k+1)/2}\,_{2}F_{1}\left(b,\frac{k+1}{2},c;x\right)$

We make two changes of variable selectively, $a=(k+1)/2$ and $z=k/2$, such
that

$\displaystyle\int_{0}^{\infty}ds\,M\left(b,c;xzs^{2}\right)\,s^{2a-1}\,e^{-zs^{2}}$
$\displaystyle=\frac{1}{2}\Gamma(a)\,z^{-a}\,\,_{2}F_{1}\left(b,a,c;x\right)$

Lastly, let $t=s^{2}$, and $x=q/z$, we get

$\displaystyle\int_{0}^{\infty}dt\,M\left(b,c;qt\right)\,t^{a-1}\,e^{-zt}$
$\displaystyle=\Gamma(a)\,z^{-a}\,\,_{2}F_{1}\left(b,a,c;q/z\right)$

which is the Laplace transform of the Kummer function mentioned in 13.10.3 of
DLMF. Therefore, this Lemma is essentially a disguised Laplace transform.

∎

It is then straightforward to obtain the known expression of the CDF of
Student’s t as777 See also Wikipedia at
https://en.wikipedia.org/wiki/Student%27s_t-
distribution#Cumulative_distribution_function

(87) $\displaystyle\Phi[L_{1,k}](x)$
$\displaystyle=\frac{1}{2}+\frac{x}{\sqrt{k}}\,\frac{1}{B(\frac{k}{2},\frac{1}{2})}\,_{2}F_{1}\left(\frac{1}{2},\frac{k+1}{2};\frac{3}{2};-\frac{x^{2}}{k}\right)$
$\displaystyle(x^{2}<k)$

Next, the CDF of Student’s t can also be expressed in the regularized
incomplete beta function $I_{p}(a,b)$, where $0\leq p\leq 1$ (See 8.17.2 in
DLMF) such that

(88) $\displaystyle\Phi[L_{1,k}](x)$
$\displaystyle=1-\frac{1}{2}I_{p}\left(\frac{k}{2},\frac{1}{2}\right)=\frac{1}{2}+\frac{1}{2}I_{q}\left(\frac{1}{2},\frac{k}{2}\right)$
$\displaystyle(x\geq 0)$
$\displaystyle\quad\,\,\text{where}\,\,p=\frac{k}{x^{2}+k},q=1-p.$

This is an application of the following equation (by combining 8.17.7 and
15.8.1 in DLMF):

$\displaystyle I_{q}(a,b)$
$\displaystyle=\frac{1}{aB(a,b)}\,\left(\frac{q}{p}\right)^{a}\,\,_{2}F_{1}\left(a,a+b,a+1;-\frac{q}{p}\right)$

where we assign $a=1/2$, $b=k/2$, and $q/p=x^{2}/k$.

###  8.6. GSaS Characteristic Function

We explore the CF of GSaS that was mentioned in (33):

$\displaystyle\phi_{\alpha,k}(\zeta)$
$\displaystyle=\sqrt{2\pi}\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{\zeta}{s}\right)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle(k>0)$

It forms a high transcendental function, fractionally extended such that it
subsumes both the stretched exponential function $e^{-\zeta^{\alpha}}$ in
(89), and the _modified Bessel function of the second kind_ 888 See also
Wikipedia at
https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions.
$K_{\frac{k}{2}}(\zeta\sqrt{k})$ in (90) (See scipy.special.kn and (11.117) in
[1]).

The first part is straightforward. Set $k=1$, the LHS by definition of the
$\alpha$-stable distribution is

(89) $\displaystyle\phi_{\alpha,1}(\zeta)=e^{-\zeta^{\alpha}}$
$\displaystyle=\sqrt{2\pi}\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{\zeta}{s}\right)\,\overline{\chi}_{\alpha,1}(s)$
$\displaystyle(\zeta\geq 0)$
$\displaystyle=\int_{0}^{\infty}\,\frac{ds}{s}\,\mathcal{N}\left(\frac{\zeta}{s}\right)\,\left(\sqrt{2\pi}s\,\overline{\chi}_{\alpha,1}(s)\right)$

The second line is essentially the same as the second line of (59). Note the
term in the parenthesis: $\sqrt{2\pi}s\,\overline{\chi}_{\alpha,1}(s)\propto
V_{\alpha}(s)$. And $V_{\alpha}(s)=\chi_{\alpha,2}(s)$. We can see that,
dimensionally speaking, the $s^{k}$ term is correct.

The second part is based on the CF of Student’s t, $\phi_{1,k}(\zeta)$, such
that

(90)
$\displaystyle\phi_{1,k}(\zeta)=\frac{t^{k/2}}{2^{k/2-1}\,\Gamma(\frac{k}{2})}K_{\frac{k}{2}}(t)$
$\displaystyle=\sqrt{2\pi}\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{\zeta}{s}\right)\,\overline{\chi}_{1,k}(s),\quad\,\,\text{where}\,\,t=\zeta\sqrt{k},\zeta\geq
0$

The complex normalization constant between $\phi_{1,k}(\zeta)$ and
$K_{\frac{k}{2}}(t)$ is to ensure the general requirement of a CF such that
$\phi_{\alpha,k}(0)=1$. The $t^{k/2}$ term is due to the divergent asymptotic
of $K_{\frac{k}{2}}(t)$ at $t=0$ (See (11.124) in [1]):

$\displaystyle\lim_{t\to 0}K_{\nu}(t)$
$\displaystyle=2^{\nu-1}\,\Gamma(\nu)\,t^{-\nu},\quad\nu=\frac{k}{2}>0$

###  8.7. GSaS Series Representation of PDF and Tail Behavior

The PDF of GSaS $(k>0)$ has two known series representations. One in term of
$x$ and the other in terms of $1/x$. Both have issues in terms of radius of
convergence. More future work is needed in this area.

Here we present the $1/x$ result to investigate the tail behavior. This is a
working example of the four-parameter Wright function (126). (The small $x$
series is presented in Section A.1.)

######  Lemma 8.5.

The series representation of GSaS PDF in terms of $1/x$ is

(91) $\displaystyle L_{\alpha,k}({x})$
$\displaystyle=\,\left(\frac{S_{\alpha,k}}{2\sqrt{\pi}}\right)\,\frac{1}{x}{\left(\frac{x}{\widetilde{\sigma}_{\alpha,k}}\right)}^{-k+1}\,W\left[\begin{matrix}\frac{\alpha}{2},&\frac{k}{2}\\\
-\frac{\alpha}{2},&0\end{matrix}\right]\left(-\left(\frac{x}{\widetilde{\sigma}_{\alpha,k}}\right)^{-\alpha}\right)$
$\displaystyle=\,\left(\frac{S_{\alpha,k}}{2\sqrt{\pi}\,\widetilde{\sigma}_{\alpha,k}}\right){\left(\frac{x}{\widetilde{\sigma}_{\alpha,k}}\right)}^{-k}\,\sum_{n=1}^{\infty}\frac{\Gamma\left(\frac{\alpha
n}{2}+\frac{k}{2}\right)}{\Gamma(n+1)\,\Gamma(-\frac{\alpha
n}{2})}\left[-\left(\frac{x}{\widetilde{\sigma}_{\alpha,k}}\right)^{-\alpha}\right]^{n}$
(92)
$\displaystyle\,\,\text{where}\,\,S_{\alpha,k}:=\frac{\alpha\Gamma(\frac{k-1}{2})}{\Gamma(\frac{k-1}{\alpha})}\,\,\text{and}\,\,\widetilde{\sigma}_{\alpha,k}:=\frac{\sqrt{2}}{{\sigma}_{\alpha,k}}=2\big{/}\left(k^{1/2-1/\alpha}\right)$

This series works better for smaller $\alpha$ where the Pareto tails are more
prominent.

###### Proof.

From (69), we get

$\displaystyle L_{\alpha,k}({x})$
$\displaystyle=S_{\alpha,k}\int_{0}^{\infty}\,ds\,\mathcal{N}(xs)\,{\left(\frac{s}{{\sigma}_{\alpha,k}}\right)}^{k-1}\,W_{\frac{\alpha}{2},0}\left(-{\left(\frac{s}{{\sigma}_{\alpha,k}}\right)}^{\alpha}\right)$
$\displaystyle=S_{\alpha,k}{\sigma}_{\alpha,k}\,\int_{0}^{\infty}\,dz\,\mathcal{N}(xz\,{\sigma}_{\alpha,k})\,{z}^{k-1}\,W_{\frac{\alpha}{2},0}\left(-{z}^{\alpha}\right)$

where $z=s/{\sigma}_{\alpha,k}$.

Substitute the Wright function with its series representation,

$\displaystyle L_{\alpha,k}({x})$
$\displaystyle=\frac{\sqrt{2}S_{\alpha,k}}{\widetilde{\sigma}_{\alpha,k}}\,\,\int_{0}^{\infty}dz\,\left(\frac{1}{\sqrt{2\pi}}e^{-{(zx/\widetilde{\sigma}_{\alpha,k})}^{2}}\right)\,\sum_{n=1}^{\infty}\frac{{(-1)}^{n}{z}^{\alpha
n+k-1}}{n!\,\Gamma(-\frac{\alpha n}{2})}$

Integrate term by term, each integral is just the moment formula of a normal
distribution.

$\displaystyle L_{\alpha,k}({x})$
$\displaystyle=\frac{S_{\alpha,k}}{\sqrt{\pi}\widetilde{\sigma}_{\alpha,k}}\,\,\sum_{n=1}^{\infty}\frac{1}{2}\frac{{(-1)}^{n}}{\Gamma(n+1)\,\Gamma(-\frac{\alpha
n}{2})}\,\Gamma\left(\frac{\alpha
n+k}{2}\right){\left(\frac{\widetilde{\sigma}_{\alpha,k}}{x}\right)}^{\alpha
n+k}$

Re-arrange the term inside the summation, we arrive at the second line of
(91). Then replace the summation with the four-parameter Wright function
(126). ∎

Take the first term of the summation, we observe that the tail behavior of
GSaS is proportional to $x^{-(\alpha+k)}$. This illustrates the effect of $k$
in which larger $k$ makes the tail lighter.

To validate the Pareto tail behavior, when $k\to 1$, it becomes $\alpha
c_{\alpha}x^{-(1+\alpha)}$, where
$c_{\alpha}=\Gamma(\alpha)\sin(\frac{\alpha\pi}{2})/\pi$, consistent with
(3.43) of Nolan (2020)[23].

##  9\. GEP: Generalized Exponential Power Distribution

The exponential power distribution
$\mathcal{E}_{\alpha}(x)=e^{-{|x|}^{\alpha}}/2\Gamma\left(\frac{1}{\alpha}+1\right)$
is mentioned in Definition 3.6. It has been used to model asset returns in
finance, as shown in the LHS of (59). Its usefulness stems from its
simplicity. Unlike its $\alpha$-stable counterpart, the existence of moments
is never an issue.

GEP $\mathcal{E}_{\alpha,k}$ enriches $\mathcal{E}_{\alpha}$ with the _degree
of freedom_ parameter $k$. We have shown that
$\mathcal{E}_{\alpha,k}=L_{\alpha,-k}$ by the clever design of FCM. Therefore
its properties can be obtained indirectly from those of GSaS. We still carry
out some proofs here based on its product distribution definition (34), for
completeness sake as well as for validation.

Its peak PDF is simply
$\mathcal{E}_{\alpha,k}(0)=\frac{1}{\sqrt{2\pi}}/\mathbb{E}(X|{\overline{\chi}_{\alpha,k}})$.
It can be standardized by multiplying with its standard deviation, which is
derived below.

All the moments of $\mathcal{E}_{\alpha,k}$ exist. The reader can take a
closer look at Figure 6. All the kurtosis contours are continuous on the half
plane of $k>0$. Its moment formula is straightforward:

######  Lemma 9.1.

The $n$-th moment of $\mathcal{E}_{\alpha,k}$ is

(93) $\displaystyle\mathbb{E}(X^{n}|\mathcal{E}_{\alpha,k})$
$\displaystyle=\frac{\mathbb{E}(X^{n}|\mathcal{N})\,\mathbb{E}(X^{n+1}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}$

where $n$ is even. All the odd moments are zero.

###### Proof.

The integral of the $n$-th moment from (34) is

$\displaystyle\mathbb{E}(X^{n}|\mathcal{E}_{\alpha,k})$
$\displaystyle=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\,\int_{-\infty}^{\infty}x^{n}\,dx\,\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{x}{s}\right)\,\overline{\chi}_{\alpha,k}(s)$

With a change of variable $z=x/s$, we can separate the two integrals:

$\displaystyle\mathbb{E}(X^{n}|\mathcal{E}_{\alpha,k})$
$\displaystyle=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\,\int_{-\infty}^{\infty}z^{n}\mathcal{N}\left(z\right)\,dz\,\int_{0}^{\infty}\,s^{n+1}\,\overline{\chi}_{\alpha,k}(s)\,ds$
$\displaystyle=\frac{\mathbb{E}(X^{n}|\mathcal{N})\,\mathbb{E}(X^{n+1}|\overline{\chi}_{\alpha,k})}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}$

∎

The excess kurtosis of $\mathcal{E}_{\alpha,k}$ in (38) is derived from
$\mathbb{E}(X^{4}|\mathcal{E}_{\alpha,k})/\mathbb{E}(X^{2}|\mathcal{E}_{\alpha,k})^{2}$.

The CDF of $\mathcal{E}_{\alpha,k}$ is derived below by following the same
logic in Section 8.4.

######  Lemma 9.2.

The CDF of $\mathcal{E}_{\alpha,k}$ is

(94) $\displaystyle\Phi[\mathcal{E}_{\alpha,k}](x)$
$\displaystyle:=\int_{-\infty}^{x}\,\mathcal{E}_{\alpha,k}\left(z\right)dz=\frac{1}{2}+\frac{1}{2\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\int_{0}^{\infty}\,s\,ds\,\operatorname{erf}\left(\frac{x}{\sqrt{2}\,s}\right)\,\overline{\chi}_{\alpha,k}(s)$

###### Proof.

We begin with the double integral via the PDF, such that

$\displaystyle\Phi[\mathcal{E}_{\alpha,k}](x)$
$\displaystyle=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\int_{-\infty}^{x}\,dz\int_{0}^{\infty}\,ds\,\mathcal{N}\left(\frac{z}{s}\right)\,\overline{\chi}_{\alpha,k}(s)$

The order of the two integrals are switched, and we make a change of variable
$u=z/s$:

$\displaystyle\Phi[\mathcal{E}_{\alpha,k}](x)$
$\displaystyle=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\int_{0}^{\infty}\,ds\,\left(\int_{-\infty}^{x}\,dz\,\mathcal{N}\left(\frac{z}{s}\right)\right)\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\int_{0}^{\infty}\,s\,ds\,\left(\int_{-\infty}^{x/s}\,du\,\mathcal{N}(u)\right)\,\overline{\chi}_{\alpha,k}(s)$

The inner integral is replaced with the error function:

$\displaystyle\Phi[\mathcal{E}_{\alpha,k}](x)$
$\displaystyle=\frac{1}{\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\int_{0}^{\infty}\,s\,ds\,\left[\frac{1}{2}+\frac{1}{2}\operatorname{erf}\left(\frac{x}{\sqrt{2}\,s}\right)\right]\,\overline{\chi}_{\alpha,k}(s)$
$\displaystyle=\frac{1}{2}+\frac{1}{2\mathbb{E}(X|\overline{\chi}_{\alpha,k})}\int_{0}^{\infty}\,s\,ds\,\operatorname{erf}\left(\frac{x}{\sqrt{2}\,s}\right)\,\overline{\chi}_{\alpha,k}(s)$

We arrive at the desired formula for the CDF. ∎

##  10\. GAS: Generalized Alpha-Stable Distribution

The invention of GAS is experimental. Since GSaS is very successful, it is a
natural next step to add skewness to it.

We take a simple approach – The separation of concerns between the skew
Gaussian kernel $g_{\alpha}^{\theta}(x,s)$ and fractional chi-1 in
$L_{\alpha}^{\theta}$ is extended along the FCM side, as shown below from the
first line to the second line:

$\displaystyle L_{\alpha}^{\theta}({x})=L_{\alpha,1}^{\theta}(x)$
$\displaystyle=\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\overline{\chi}_{\alpha,1}(s)$
(95) $\displaystyle\implies\quad\quad L_{\alpha,k}^{\theta}({x})$
$\displaystyle:=\int_{0}^{\infty}s\,ds\,g_{\alpha}^{\theta}(x,s)\,\overline{\chi}_{\alpha,k}({s})$

The skewness of GAS comes from $g_{\alpha}^{\theta}(x,s)$ when $\theta\neq 0$.
When $\theta=0$, $g_{\alpha}^{\theta}(x,s)=\mathcal{N}(xs)$ and GAS becomes
GSaS.

Analyzing the behavior of skewness in depth when $|k|>1$ is a complicated
topic. $\theta$ is confined by $L_{\alpha,k}^{\theta}({x})>0$ for all
$x\in\mathbb{R}$. When $|\theta|$ is small, the skew PDF behaves well. But the
valid range of $\theta$ appears to be _much smaller_ than the _Feller-Takayasu
diamond_ : $\theta\leq\min\\{\alpha,2-\alpha\\}$ [20, 19]. The larger $k$ is,
the smaller range of $\theta$. This issue is left for future research.

######  Lemma 10.1.

(GAS Symmetry Relation) The following symmetry relation holds, which is an
extension from $\alpha$-stable (See Section 4 in [20]):

$\displaystyle L_{\alpha,k}^{\theta}({-x})$
$\displaystyle=L_{\alpha,k}^{-\theta}(x)$

Therefore, one only needs to consider the case of $x>0$.

###### Proof.

Both $\alpha,\theta$ only show up in the skew kernel, not in FCM. We just need
to prove the relation: $g_{\alpha}^{\theta}(-x,s)=g_{\alpha}^{-\theta}(x,s)$.

This is easy to show since the cosine function is symmetric and the tangent
function is asymmetric. In (30), when $x$ and $\tau$ change sign
simultaneously, the term
$\cos\left(\tau\,{(st)}^{\alpha}+\frac{x}{q}\,st\right)$ remains the same.

∎

Two subjects are unresolved for GAS as of this writing. First, the analytic
form for the moments of GAS is wanting. Second, the $k$ extension of a well-
known reciprocal relation between $\alpha$ and $1/\alpha$ is unknown yet. When
$k=1$, Feller first showed

$\displaystyle\frac{1}{x^{\alpha+1}}\,L_{1/\alpha}^{\theta}(x^{-\alpha})$