Discrete-stable distribution

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 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^[2] or even semantic networks.^[3]

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 the Poisson distribution.^[4] It is the only discrete-stable distribution for which the mean and all higher-order moments are finite.^{[dubious – discuss]}

Definition

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

G (s | ν, a) = \sum_{n = 0}^{\infty} P (N | ν, a) (1 - s)^{N} = \exp (- a s^{ν}) .

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

lim_{N \to \infty} P (N | ν, a) \sim \frac{1}{N^{ν + 1}} .

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

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

φ (t; a, ν) = \exp [a {(e^{i t} - 1)}^{ν}]

, with

a > 0

and

0 < ν \leq 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) = {\frac{(- 1)^{N}}{N!} \frac{d^{N} G (s | ν, a)}{d s^{N}} |}_{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) = \frac{a^{N} e^{- a}}{N!} .

Expressions exist, however, that use 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 distribution 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 the stability parameter $0 < α < 1$ and scale parameter $c$ , the resultant distribution is^[9] discrete-stable with index $ν = α$ and scale parameter $a = c \sec (α π / 2)$ .

Formally, this is written

P (N | α, c \sec (α π / 2)) = \int_{0}^{\infty} P (N | 1, λ) p (λ; α, 1, c, 0) d λ

where $p (x; α, 1, c, 0)$ is the pdf of a one-sided continuous-stable distribution with symmetry parameter $β = 1$ 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 $ν \cdot α$ and reduces the power-law index of the original distribution by a factor of $α$ .

In other words,

P (N | ν \cdot α, c \sec (π α / 2)) = \int_{0}^{\infty} P (N | α, λ) p (λ; ν, 1, c, 0) d λ .

Poisson limit

In the limit $ν \to 1$ , the discrete-stable distributions behave^[9] like a Poisson distribution with mean $a \sec (ν π / 2)$ for small $N$ , but for $N ≫ 1$ , the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails $P (N) \sim 1 / N^{1 + ν}$ to a discrete-stable distribution is extraordinarily slow^[10] when $ν \approx 1$ , the limit being the Poisson distribution when $ν > 1$ and $P (N | ν, a)$ when $ν \leq 1$ .

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.
↑ Renshaw, Eric (2015-03-19). Stochastic Population Processes: Analysis, Approximations, Simulations. OUP Oxford. ISBN 978-0-19-106039-7.
↑ 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.
↑ Slamova, Lenka; Klebanov, Lev. "Modeling financial returns by discrete stable distributions" (PDF). International Conference Mathematical Methods in Economics. Retrieved 2023-07-07.
↑ 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.
1 2 Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.
1 2 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.

Discrete-stable distribution

Definition

As compound probability distributions

Poisson limit

See also

References

Further reading