Discrete-stable distribution

This article's factual accuracy is disputed. Please help to ensure that disputed statements are reliably sourced. (December 2016) (Learn how and when to remove this template message)

|

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.[dubious - discuss]

## Contents

* 1 Definition
* 2 As compound probability distributions
* 3 In the Poisson limit
* 4 See also
* 5 References
* 6 Further reading

## Definition

The discrete-stable distributions are defined[5] through their probability-generating function

G(s|ν,a)=∑n=0∞P(N|ν,a)(1−s)N=exp⁡(−asν).

In the above, a>0 is a scale parameter and 0<ν≤1 describes the power-law behaviour such that when 0<ν<1,

limN->∞P(N|ν,a)∼1Nν+1.

When ν=1 the distribution becomes the familiar Poisson distribution with mean a.

The characteristic function of a discrete-stable distribution has the form:[6]

φ(t;a,ν)=exp⁡[a(eit−1)ν], with a>0 and 0<ν≤1.

Again, when ν=1 the distribution becomes the Poisson distribution with mean a.

The original distribution is recovered through repeated differentiation of the generating function:

P(N|ν,a)=(−1)NN!dNG(s|ν,a)dsN|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|ν=1,a)=aNe−aN!.

Expressions do exist, however, using special functions for the case ν=1/2[7] (in terms of Bessel functions) and ν=1/3[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, λ, 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<α<1 and scale parameter c the resultant distribution is[9] discrete-stable with index ν=α and scale parameter a=csec⁡(απ/2).

Formally, this is written:

P(N|α,csec⁡(απ/2))=∫0∞P(N|1,λ)p(λ;α,1,c,0)dλ

where p(x;α,1,c,0) is the pdf of a one-sided continuous-stable distribution with symmetry paramètre β=1 and location parameter μ=0.

A more general result[8] states that forming a compound distribution from any discrete-stable distribution with index ν with a one-sided continuous-stable distribution with index α results in a discrete-stable distribution with index ν⋅α, reducing the power-law index of the original distribution by a factor of α.

In other words,

P(N|ν⋅α,csec⁡(πα/2))=∫0∞P(N|α,λ)p(λ;ν,1,c,0)dλ.

## In the Poisson limit

In the limit ν->1, the discrete-stable distributions behave[9] like a Poisson distribution with mean asec⁡(νπ/2) for small N, however for N≫1, the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails P(N)∼1/N1+ν to a discrete-stable distribution is extraordinarily slow[10] when ν≈1 \- the limit being the Poisson distribution when ν>1 and P(N|ν,a) when ν≤1.

## See also

* Stable distribution
* Poisson distribution

## References

1. ↑ Steutel, F. W.; van Harn, K. (1979). "Discrete Analogues of Self-Decomposability and Stability". Annals of Probability 7 (5): 893–899. doi:10.1214/aop/1176994950. https://pure.tue.nl/ws/files/1956807/Metis199408.pdf.
2. ↑ 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.
3. ↑ 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. doi:10.1207/s15516709cog2901_3. PMID 21702767.
4. ↑ Renshaw, Eric (2015-03-19) (in en). Stochastic Population Processes: Analysis, Approximations, Simulations. OUP Oxford. ISBN 978-0-19-106039-7. https://books.google.com/books?id=pqE1CgAAQBAJ&pg=PA134.
5. ↑ 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. doi:10.1088/0305-4470/35/49/101. Bibcode: 2002JPhA...35L.745H.
6. ↑ "Modeling financial returns by discrete stable distributions". International Conference Mathematical Methods in Economics. http://mme2012.opf.slu.cz/proceedings/pdf/138_Slamova.pdf. Retrieved 2023-07-07.
7. ↑ 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. doi:10.1088/0305-4470/36/46/004. Bibcode: 2003JPhA...3611585M.
8. ↑ 8.0 8.1 Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.
9. ↑ 9.0 9.1 Lee, W. H.; Hopcraft, K. I.; Jakeman, E. (2008). "Continuous and discrete stable processes". Physical Review E 77 (1): 011109–1 to 011109–04. doi:10.1103/PhysRevE.77.011109. PMID 18351820. Bibcode: 2008PhRvE..77a1109L.
10. ↑ 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. doi:10.1088/0305-4470/37/48/L01. Bibcode: 2004JPhA...37L.635H.

## 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. https://archive.org/details/limitdistributio00gned_0.
* Ibragimov, I.; Linnik, Yu (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Discrete-stable distribution. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Discrete-stable_distribution&oldid=218130"