📂Mathematical Statistics

Stable Distribution in Probability Theory

Definition ¹

Implicit definition

For a random variable $X$ that is not a degenerate distribution, let $X_{1} , \cdots , X_{n} \overset{\text{iid}}{\sim} X$. For all $n > 1$, the existence of $c_{n} > 0$ and $d_{n} \in \mathbb{R}$ satisfying the following is a necessary and sufficient condition that $X$ is stable. $$ X_{1} + \cdots + X_{n} \sim c_{n} X + d_{n} $$

Explicit definition

For $0 < \alpha \le 2$ and $-1 \le \beta \le 1$, let the characteristic function of the random variable $Z$ be given by the following. $$ \varphi_{Z} (u) = \begin{cases} \exp \left( - \left| u \right|^{\alpha} \left[ 1 - i \beta \sign u \tan \left( \pi \alpha / 2 \right) \right] \right) & , \text{if } \alpha \ne 1 \\ \exp \left( - \left| u \right| \left[ 1 + i \beta \left( 2 / \pi \right) \sign u \log \left| u \right| \right] \right) & , \text{if } \alpha = 1 \end{cases} $$ For this, the existence of $a > 0$ and $b \in \mathbb{R}$ satisfying $X \sim a Z + b$ is a necessary and sufficient condition that $X$ is stable. Ironically, despite being called the explicit definition, the probability density function of a stable distribution admits a closed form only in special cases.

Parameterization

A stable law is denoted $S \left( \alpha , \beta , \gamma, \delta ; k \right)$ to include four parameters $\alpha, \beta, \gamma, \delta$ and an integer $k$ specifying the form. Nolan, in his book, presents ten other parameterizations in addition to the explicit definition above and urges that literature referring to stable distributions should clearly state which parameterization is used. We consider two of them here.

• $\alpha \in ( 0 , 2 ]$: index of stability
• $\beta \in [ -1 , 1 ]$: skewness
• $\gamma > 0$: scale
• $\delta \in \mathbb{R}$: location
• $k = 0 , \cdots , 10$: form

Nolan’s form 0

If the random variable $X$ satisfies the following, it is called $X \sim S \left( \alpha , \beta , \gamma, \delta ; 0 \right)$. $$ X \sim \begin{cases} \gamma \left( Z - \beta \tan \left( \pi \alpha / 2 \right) \right) & , \text{if } \alpha \ne 1 \\ \gamma Z + \delta & , \text{if } \alpha = 1 \end{cases} $$

Nolan’s form 1

If the random variable $X$ satisfies the following, it is called $X \sim S \left( \alpha , \beta , \gamma, \delta ; 1 \right)$. $$ X \sim \begin{cases} \gamma Z + \delta & , \text{if } \alpha \ne 1 \\ \gamma Z + \left( \delta + \beta \left( 2 / \pi \right) \gamma \log \gamma \right) & , \text{if } \alpha = 1 \end{cases} $$

Explanation

Stable distributions are also called Lévy $\alpha$-distributions after Paul Lévy. In the implicit definition, the reason stable distributions are called “stable” can be understood as the sense of being closed under addition.

Focusing on relationships among distributions, the stable distribution is a generalization of the normal distribution, the Cauchy distribution, and the Lévy distribution: when $\alpha = 2, \beta 0$ it is the normal distribution, when $\alpha = 1, \beta = 0$ the Cauchy distribution, and when $\alpha = 1/2, \beta = 1$ the Lévy distribution. Unlike the other two, the Lévy distribution may be somewhat unfamiliar in the context of statistics; Lévy is associated with many concepts—for example, a stochastic process whose increments $X_{t+1} - X_{t}$ follow a heavy-tailed stable distribution is called a Lévy flight.

Generalized central limit theorem

There is also a generalization of the central limit theorem for stable distributions. $$ a_{n} \left( X_{1} + \cdots + X_{n} \right) - b_{n} \overset{D}{\to} Z $$ For $X_{1} , \cdots , X_{n} \overset{\text{iid}}{\sim} X$, we define $DA(Z)$ as: the existence of $a_{n} > 0$ and $b_{n} \in \mathbb{R}$ satisfying the above, which is a necessary and sufficient condition that $X$ belongs to the domain of attraction $DA \left( Z \right)$.

The generalized central limit theorem states here that the necessary and sufficient condition for $Z$ to be a stable distribution $0 < \alpha \le 2$ is that $X$ belongs to $DA(Z)$.