Discrete-stable distribution

The discrete-stable distributions[1] are a class of probability distribution and 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 classes of 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.

The distribution

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

Q(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\le1 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^NQ(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 compound probability distributions where the mean, a, of a Poisson distribution is defined as 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(\pi \alpha / 2).

Formally, this is written:

P(N, \alpha, c \sec(\pi \alpha / 2)) = \int_0^\infty P(N, 1, a)p(a, \alpha, 1, c, 0) \, da

where p(a, \alpha, 1, c, 0) is the pdf of a one-sided continuous-stable distribution with symmetry parameter \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,

P(N, \nu \cdot \alpha, c \sec(\pi \alpha / 2) = \int_0^\infty P(N, \alpha, a)p(a, \nu, 1, c, 0) \, da.

In the Poisson limit

In the limit \nu \rarr 1, the discrete-stable distributions behave[7] like a Poisson distribution with mean a \sec(\pi \nu / 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

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.
  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. ↑ 6.0 6.1 Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.
  7. ↑ 7.0 7.1 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.
  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