Discrete-stable distribution

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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[dubious ].

Definition[edit]

The discrete-stable distributions are defined[4] through their probability-generating function

In the above, is a scale parameter and describes the power-law behaviour such that when ,

When the distribution becomes the familiar Poisson distribution with mean .

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

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

Expressions do exist, however, using special functions for the case [5] (in terms of Bessel functions) and [6] (in terms of hypergeometric functions).

As compound probability distributions[edit]

The entire class of discrete-stable distributions can be formed as Poisson compound probability distributions where the mean, , 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 and scale parameter the resultant distribution is[7] discrete-stable with index and scale parameter .

Formally, this is written:

where is the pdf of a one-sided continuous-stable distribution with symmetry paramètre and location parameter .

A more general result[6] states that forming a compound distribution from any discrete-stable distribution with index with a one-sided continuous-stable distribution with index results in a discrete-stable distribution with index , reducing the power-law index of the original distribution by a factor of .

In other words,

In the Poisson limit[edit]

In the limit , the discrete-stable distributions behave[7] like a Poisson distribution with mean for small , however for , the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails to a discrete-stable distribution is extraordinarily slow[8] when - the limit being the Poisson distribution when and when .

See also[edit]

References[edit]

  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. 
  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. 
  4. ^ Hopcraft, K. I.; Jakeman, E.; Matthews, J. O. (2002). "Generation and monitoring of a discrete stable random process". Journal of Physics A: General Physics. 35 (49): L745–752. doi:10.1088/0305-4470/35/49/101. 
  5. ^ 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: Mathematical and General. 36: 11585–11603. doi:10.1088/0305-4470/36/46/004. 
  6. ^ a b Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham. 
  7. ^ a b 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. 
  8. ^ Hopcraft, K. I.; Jakeman, E.; Matthews, J. O. (2004). "Discrete scale-free distributions and associated limit theorems". Journal of Physics A: General Physics. 37 (48): L635–L642. doi:10.1088/0305-4470/37/48/L01. 

Further reading[edit]

  • 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.