Английская Википедия:Discrete-stable distribution
Шаблон:Disputed The discrete-stable distributions[1] are a class of probability distributions with the property that the sum of several random variables from such a distribution under appropriate scaling is distributed according to the same family. They are the discrete analogue of the continuous-stable distributions.
The discrete-stable distributions have been used in numerous fields, in particular in scale-free networks such as the internet, social networks[2] or even semantic networks.[3]
Both the discrete and continuous classes of stable distribution have properties such as infinite divisibility, power law tails and unimodality.
The most well-known discrete stable distribution is the Poisson distribution which is a special case.[4] It is the only discrete-stable distribution for which the mean and all higher-order moments are finite.Шаблон:Dubious
Definition
The discrete-stable distributions are defined[5] through their probability-generating function
- <math>G(s| \nu,a)=\sum_{n=0}^\infty P(N| \nu,a)(1-s)^N = \exp(-a s^\nu).</math>
In the above, <math>a>0</math> is a scale parameter and <math>0<\nu\le1</math> describes the power-law behaviour such that when <math>0<\nu<1</math>,
- <math> \lim_{N \to \infty}P(N|\nu,a) \sim \frac{1}{N^{\nu+1}}.</math>
When <math>\nu=1</math> the distribution becomes the familiar Poisson distribution with mean <math>a</math>.
The characteristic function of a discrete-stable distribution has the form:[6]
- <math> \varphi(t; a, \nu) = \exp \left[a \left( e^{it} - 1 \right)^\nu \right]</math>, with <math>a>0</math> and <math>0<\nu\le1</math>.
Again, when <math>\nu=1</math> the distribution becomes the Poisson distribution with mean <math>a</math>.
The original distribution is recovered through repeated differentiation of the generating function:
- <math>P(N|\nu,a)= \left.\frac{(-1)^N}{N!}\frac{d^NG(s|\nu,a)}{ds^N}\right|_{s=1}.</math>
A closed-form expression using elementary functions for the probability distribution of the discrete-stable distributions is not known except for in the Poisson case, in which
- <math>\!P(N| \nu=1, a)= \frac{a^N e^{-a}}{N!}.</math>
Expressions do exist, however, using special functions for the case <math>\nu=1/2</math>[7] (in terms of Bessel functions) and <math>\nu=1/3</math>[8] (in terms of hypergeometric functions).
As compound probability distributions
The entire class of discrete-stable distributions can be formed as Poisson compound probability distributions where the mean, <math>\lambda</math>, of a Poisson distribution is defined as a random variable with a probability density function (PDF). When the PDF of the mean is a one-sided continuous-stable distribution with stability parameter <math>0 < \alpha < 1</math> and scale parameter <math>c</math> the resultant distribution is[9] discrete-stable with index <math>\nu = \alpha</math> and scale parameter <math>a = c \sec( \alpha \pi / 2)</math>.
Formally, this is written:
- <math>
P(N| \alpha, c \sec( \alpha \pi / 2)) = \int_0^\infty P(N| 1, \lambda)p(\lambda; \alpha, 1, c, 0) \, d\lambda </math>
where <math>p(x; \alpha, 1, c, 0)</math> is the pdf of a one-sided continuous-stable distribution with symmetry paramètre <math>\beta=1</math> and location parameter <math>\mu = 0</math>.
A more general result[8] states that forming a compound distribution from any discrete-stable distribution with index <math>\nu</math> with a one-sided continuous-stable distribution with index <math>\alpha</math> results in a discrete-stable distribution with index <math>\nu \cdot \alpha</math>, reducing the power-law index of the original distribution by a factor of <math>\alpha</math>.
In other words,
- <math>
P(N| \nu \cdot \alpha, c \sec(\pi \alpha / 2)) = \int_0^\infty P(N| \alpha, \lambda)p(\lambda; \nu, 1, c, 0) \, d\lambda. </math>
In the Poisson limit
In the limit <math>\nu \rarr 1</math>, the discrete-stable distributions behave[9] like a Poisson distribution with mean <math>a \sec(\nu \pi / 2)</math> for small <math>N</math>, however for <math>N \gg 1</math>, the power-law tail dominates.
The convergence of i.i.d. random variates with power-law tails <math>P(N) \sim 1/N^{1 + \nu}</math> to a discrete-stable distribution is extraordinarily slow[10] when <math>\nu \approx 1</math> - the limit being the Poisson distribution when <math>\nu > 1</math> and <math>P(N| \nu, a)</math> when <math>\nu \leq 1</math>.
See also
References
Further reading
- Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley. Шаблон:ISBN
- Шаблон:Cite book
- Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Barabási, Albert-László (2003). Linked: how everything is connected to everything else and what it means for business, science, and everyday life. New York, NY: Plum.
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ 8,0 8,1 Шаблон:Cite thesis
- ↑ 9,0 9,1 Шаблон:Cite journal
- ↑ Шаблон:Cite journal