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 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 infinitely divisibility, power law tails and unimodality.
The most well-known discrete stable distribution is the Poisson distribution which is a special case as the only discrete-stable distribution for which the mean and all higher-order moments are finite.
Definition
The discrete-stable distributions are defined[4] through their probability-generating function
\( G(s| \nu,a)=\sum_{n=0}^\infty P(N| \nu,a)(1-s)^N = \exp(-a s^\nu). \)
In the above, a>0 is a scale parameter and \( 0<\nu \leq 1 \) describes the power-law behaviour such that when \( 0<\nu <1, \)
\( \lim_{N \to \infty}P(N|\nu,a) \sim \frac{1}{N^{\nu+1}}. \)
When \( \nu =1 \) the distribution becomes the familiar Poisson distribution with mean a.
The original distribution is recovered through repeated differentiation of the generating function:
\( P(N|\nu,a)= \left.\frac{(-1)^N}{N!}\frac{d^NG(s|\nu,a)}{ds^N}\right|_{s=1}. \)
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
\( \!P(N| \nu=1, a)= \frac{a^N e^{-a}}{N!}. \)
Expressions do exist, however, using special functions for the case \( \nu =1/2 \) [5] (in terms of Bessel functions) and \( \nu=1/3 \)[6] (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, λ {\displaystyle \lambda } \lambda , 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 \( 0<\alpha <1 \) and scale parameter c the resultant distribution is[7] discrete-stable with index \( \nu =\alpha \) and scale parameter \( a = c \sec( \alpha \pi / 2). \)
Formally, this is written:
\( P(N| \alpha, c \sec( \alpha \pi / 2)) = \int_0^\infty P(N| 1, \lambda)p(\lambda; \alpha, 1, c, 0) \, d\lambda \)
where \( p(x; \alpha, 1, c, 0) \) is the pdf of a one-sided continuous-stable distribution with symmetry paramètre \( \beta =1 \) and location parameter \( \mu =0 \).
A more general result[6] states that forming a compound distribution from any discrete-stable distribution with index \( \nu \) with a one-sided continuous-stable distribution with index \( \alpha \) results in a discrete-stable distribution with index \( \nu \cdot \alpha \) , reducing the power-law index of the original distribution by a factor of \( \alpha \) .
In other words,
\( {\displaystyle P(N|\nu \cdot \alpha ,c\sec(\pi \alpha /2))=\int _{0}^{\infty }P(N|\alpha ,\lambda )p(\lambda ;\nu ,1,c,0)\,d\lambda .} \)
In the Poisson limit
In the limit \( \nu \rightarrow 1 \) , the discrete-stable distributions behave[7] like a Poisson distribution with mean \( a \sec(\nu \pi / 2)\) for small N, however for \( N\gg 1 \) , the power-law tail dominates.
The convergence of i.i.d. random variates with power-law tails \( P(N)\sim 1/N^{{1+\nu }} \) to a discrete-stable distribution is extraordinarily slow[8] when\( \nu \approx 1 \) - the limit being the Poisson distribution when \( \nu > 1 \) and \( P(N| \nu, a) \) when \( \nu \leq 1 \).
See also
Stable distribution
Poisson distribution
References
Steutel, F. W.; van Harn, K. (1979). "Discrete Analogues of Self-Decomposability and Stability" (PDF). Annals of Probability. 7 (5): 893–899. doi:10.1214/aop/1176994950.
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.
Steyvers, M.; Tenenbaum, J. B. (2005). "The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth". Cognitive Science. 29 (1): 41–78. arXiv:cond-mat/0110012. doi:10.1207/s15516709cog2901_3. PMID 21702767. S2CID 6000627.
Hopcraft, K. I.; Jakeman, E.; Matthews, J. O. (2002). "Generation and monitoring of a discrete stable random process". Journal of Physics A. 35 (49): L745–752. Bibcode:2002JPhA...35L.745H. doi:10.1088/0305-4470/35/49/101.
Matthews, J. O.; Hopcraft, K. I.; Jakeman, E. (2003). "Generation and monitoring of discrete stable random processes using multiple immigration population models". Journal of Physics A. 36 (46): 11585–11603. Bibcode:2003JPhA...3611585M. doi:10.1088/0305-4470/36/46/004.
Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.
Lee, W. H.; Hopcraft, K. I.; Jakeman, E. (2008). "Continuous and discrete stable processes". Physical Review E. 77 (1): 011109–1 to 011109–04. Bibcode:2008PhRvE..77a1109L. doi:10.1103/PhysRevE.77.011109. PMID 18351820.
Hopcraft, K. I.; Jakeman, E.; Matthews, J. O. (2004). "Discrete scale-free distributions and associated limit theorems". Journal of Physics A. 37 (48): L635–L642. Bibcode:2004JPhA...37L.635H. doi:10.1088/0305-4470/37/48/L01.
Further reading
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley. ISBN 0-471-25709-5
Gnedenko, B. V.; Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.
Ibragimov, I.; Linnik, Yu (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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