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Discrete-stable distribution

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Discrete-stable distributions 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 continuous-stable distributions.

Discrete-stable distributions have been used in numerous fields, in particular in scale-free networks such as the internet and social networks or even semantic networks.

Both 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 special case of tjhe Poisson distribution. It is the only discrete-stable distribution for which the mean and all higher-order moments are finite.

Definition

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

G ( s | ν , a ) = ∑ n = 0 ∞ P ( N | ν , a ) ( 1 − s ) N = exp ⁡ ( − a s ν ) . {\displaystyle G(s|\nu ,a)=\sum _{n=0}^{\infty }P(N|\nu ,a)(1-s)^{N}=\exp(-as^{\nu }).}

In the above, a > 0 {\displaystyle a>0} is a scale parameter and 0 < ν ≤ 1 {\displaystyle 0<\nu \leq 1} describes the power-law behaviour such that when 0 < ν < 1 {\displaystyle 0<\nu <1} ,

lim N → ∞ P ( N | ν , a ) ∼ 1 N ν + 1 . {\displaystyle \lim _{N\to \infty }P(N|\nu ,a)\sim {\frac {1}{N^{\nu +1}}}.}

When ν = 1 {\displaystyle \nu =1} , the distribution becomes the familiar Poisson distribution with the mean a {\displaystyle a} .

The characteristic function of a discrete-stable distribution has the form

φ ( t ; a , ν ) = exp ⁡ [ a ( e i t − 1 ) ν ] {\displaystyle \varphi (t;a,\nu )=\exp \left} , with a > 0 {\displaystyle a>0} and 0 < ν ≤ 1 {\displaystyle 0<\nu \leq 1} .

Again, when ν = 1 {\displaystyle \nu =1} , the distribution becomes the Poisson distribution with mean a {\displaystyle a} .

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

P ( N | ν , a ) = ( − 1 ) N N ! d N G ( s | ν , a ) d s N | s = 1 . {\displaystyle P(N|\nu ,a)=\left.{\frac {(-1)^{N}}{N!}}{\frac {d^{N}G(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 | ν = 1 , a ) = a N e − a N ! . {\displaystyle \!P(N|\nu =1,a)={\frac {a^{N}e^{-a}}{N!}}.}

Expressions exist, however, that use special functions for the case ν = 1 / 2 {\displaystyle \nu =1/2} (in terms of Bessel functions) and ν = 1 / 3 {\displaystyle \nu =1/3} (in terms of hypergeometric functions).

As compound probability distributions

The entire class of discrete-stable distributions can be formed as Poisson compound probability distribution where the mean, λ {\displaystyle \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 the stability parameter 0 < α < 1 {\displaystyle 0<\alpha <1} and scale parameter c {\displaystyle c} , the resultant distribution is discrete-stable with index ν = α {\displaystyle \nu =\alpha } and scale parameter a = c sec ⁡ ( α π / 2 ) {\displaystyle a=c\sec(\alpha \pi /2)} .

Formally, this is written

P ( N | α , c sec ⁡ ( α π / 2 ) ) = ∫ 0 ∞ P ( N | 1 , λ ) p ( λ ; α , 1 , c , 0 ) d λ {\displaystyle 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 ; α , 1 , c , 0 ) {\displaystyle p(x;\alpha ,1,c,0)} is the pdf of a one-sided continuous-stable distribution with symmetry parameter β = 1 {\displaystyle \beta =1} and location parameter μ = 0 {\displaystyle \mu =0} .

A more general result states that forming a compound distribution from any discrete-stable distribution with index ν {\displaystyle \nu } with a one-sided continuous-stable distribution with index α {\displaystyle \alpha } results in a discrete-stable distribution with index ν ⋅ α {\displaystyle \nu \cdot \alpha } and reduces the power-law index of the original distribution by a factor of α {\displaystyle \alpha } .

In other words,

P ( N | ν ⋅ α , c sec ⁡ ( π α / 2 ) ) = ∫ 0 ∞ P ( N | α , λ ) p ( λ ; ν , 1 , c , 0 ) d λ . {\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 .}

Poisson limit

In the limit ν → 1 {\displaystyle \nu \rightarrow 1} , the discrete-stable distributions behave like a Poisson distribution with mean a sec ⁡ ( ν π / 2 ) {\displaystyle a\sec(\nu \pi /2)} for small N {\displaystyle N} , but for N ≫ 1 {\displaystyle N\gg 1} , the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails P ( N ) ∼ 1 / N 1 + ν {\displaystyle P(N)\sim 1/N^{1+\nu }} to a discrete-stable distribution is extraordinarily slow when ν ≈ 1 {\displaystyle \nu \approx 1} , the limit being the Poisson distribution when ν > 1 {\displaystyle \nu >1} and P ( N | ν , a ) {\displaystyle P(N|\nu ,a)} when ν ≤ 1 {\displaystyle \nu \leq 1} .

See also

References

  1. 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.
  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. arXiv:cond-mat/0110012. doi:10.1207/s15516709cog2901_3. PMID 21702767. S2CID 6000627.
  4. Renshaw, Eric (2015-03-19). Stochastic Population Processes: Analysis, Approximations, Simulations. OUP Oxford. ISBN 978-0-19-106039-7.
  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. Bibcode:2002JPhA...35L.745H. doi:10.1088/0305-4470/35/49/101.
  6. Slamova, Lenka; Klebanov, Lev. "Modeling financial returns by discrete stable distributions" (PDF). International Conference Mathematical Methods in Economics. 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. Bibcode:2003JPhA...3611585M. doi:10.1088/0305-4470/36/46/004.
  8. ^ Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.
  9. ^ 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.
  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. Bibcode:2004JPhA...37L.635H. doi:10.1088/0305-4470/37/48/L01.

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