Fractional Poisson process

In probability theory, a fractional Poisson process is a stochastic process to model the long-memory dynamics of a stream of counts. The time interval between each pair of consecutive counts follows the non-exponential power-law distribution with parameter ν, which has physical dimension [ν] = sec^−μ, where 0 < μ ≤ 1. In other words, fractional Poisson process is non-Markov counting stochastic process which exhibits non-exponential distribution of interarrival times. The fractional Poisson process is a continuous-time process which can be thought of as natural generalization of the well-known Poisson process. Fractional Poisson probability distribution is a new member of discrete probability distributions.

The fractional Poisson process and fractional Poisson probability distribution function have been invented, developed and encouraged for applications by Nick Laskin. The terms fractional Poisson process and fractional Poisson probability distribution function were coined by Nick Laskin.

Fundamentals

The fractional Poisson probability distribution captures the long-memory effect which results in the non-exponential waiting time probability distribution function empirically observed in complex classical and quantum systems. Thus, fractional Poisson process and fractional Poisson probability distribution can be considered as natural generalization of the famous Poisson process and the Poisson probability distribution.

The idea behind the fractional Poisson process was to design counting process with non-exponential waiting time probability distribution. Mathematically the idea was realized by substitution the first-order time derivative in the Kolmogorov–Feller equation for the Poisson probability distribution function with the time derivative of fractional order.

The main outcomes are new stochastic non-Markov process – fractional Poisson process and new probability distribution function – fractional Poisson probability distribution function.

Fractional Poisson probability distribution function

The probability distribution function of fractional Poisson process is (see, Ref.[1])

$$P_\mu (n,t)=\frac{(\nu t^\mu )^n}{n!}\sum\limits_{k=0}^\infty \frac{(k+n)!}{ k!}\frac{(-\nu t^\mu )^k}{\Gamma (\mu (k+n)+1)},\qquad 0<\mu \leq 1,$$ where parameter ν has physical dimension [ν] = sec^−μ and Γ(μ(k+n)+1) is the Gamma function.

The P_μ(n,t) gives us the probability that in the time interval [0,t] we observe n events governed by fractional Poisson stream. When μ = 1 the P_μ(n,t) is transformed into the standard Poisson probability distribution, P(n,t) = P₁(n,t),

$$P(n,t)=\frac{(\overline{\nu}t )^n}{n!}\exp(-\overline{\nu} t),$$

where $\overline{\nu}$ is the rate of arrivals with physical dimension $[\overline{\nu}]=\sec ^{-1 }$.

Thus, P_μ(n,t) can be considered as fractional generalization of the standard Poisson distribution. The presence of additional parameter μ brings new features in comparison with the standard Poisson distribution.

Mean

The mean $\overline{n}_\mu$ of the fractional Poisson process is (see, Ref.[1])

$$\overline{n}_\mu =\sum\limits_{n=0}^\infty nP_\mu (n,t)=\frac{\nu t^\mu }{ \Gamma (\mu +1)}.$$

The second order moment

The second order moment of the fractional Poisson process $\overline{n^2}_\mu$ is given by (see, Ref.[1])

$$\overline{n_\mu ^2}=\sum\limits_{n=0}^\infty n^2P_\mu (n,t)=\overline{n}_\mu +\overline{n}_\mu ^2\frac{\sqrt{\pi }\Gamma (\mu +1)}{2^{2\mu -1}\Gamma (\mu +\frac 12)}.$$

Variance

The variance of the fractional Poisson process is (see, Ref.[1])

$$\sigma _\mu =\overline{n_\mu ^2}-\overline{n}_\mu ^2=\overline{n}_\mu + \overline{n}_\mu ^2\left\{ \frac{\mu B(\mu ,\frac 12)}{2^{2\mu -1}} - 1\right\},$$

where $B(\mu ,\frac 12)$ is the Beta-function.

Generating function

The generating function G_μ(s,t) for fractional Poisson probability distribution function is defined as

G_\mu (s,t)=\sum\limits_{n=0}^\infty s^nP_\mu (n,t).

The generating function of the fractional Poisson probability distribution has the following exact form (see, Ref.[1])

G_μ(s,t) = E_μ(νt^μ(s−1)),

where E_μ(z) is the Mittag-Leffler function given by its series representation

$$E_\mu (z)=\sum\limits_{m=0}^\infty \frac{z^m}{\Gamma (\mu m+1)}.$$

Moment generating function

The equation for the moment of any integer order of the fractional Poisson can be easily found by means of the moment generating function H_μ(s,t) which is defined as

$$H_\mu (s,t)=\sum\limits_{n=0}^\infty e^{-sn}P_\mu (n,t).$$

For example, for the moment of $k^{{\rm th}}$ order we have

\overline{n_\mu ^k}=(-1)^k\frac{\partial ^kH_\mu (s,t)}{\partial s^k}|_{s=0}.

The moment generating function H_μ(s,t) is (see, Ref.[1])

H_\mu (s,t)=E_\mu (\nu t^\mu (e^{-s}-1)),

or in a series form

H_\mu (s,t)=\sum\limits_{m=0}^\infty \frac 1{\Gamma (m\mu +1)}\left( \nu t^\mu (e^{-s}-1)\right) ^m,

with the help of the Mittag-Leffler function series representation.

Waiting time distribution function

A time between two successive arrivals is called as waiting time and it is a random variable. The waiting time probability distribution function is an important attribute of any arrival or counting random process.

Waiting time probability distribution function ψ_μ(τ) of the fractional Poisson process is defined as

\psi _\mu (\tau )=-\frac d{d\tau }P_\mu (\tau ),

where P_μ(τ) is the probability that a given interarrival time is greater or equal to τ

P_\mu (\tau )=1-\sum\limits_{n=1}^\infty P_\mu (n,\tau )=E_\mu (-\nu \tau^\mu ),

and P_μ(n,τ) is the fractional Poisson probability distribution function.

Thus, the waiting time probability distribution function ψ_μ(τ) of the fractional Poisson process is given by (see, Refs.[1,3]),

\psi _\mu (\tau )=\nu \tau ^{\mu -1}E_{\mu ,\mu }(-\nu \tau ^\mu ),\qquad t\geq 0,\qquad 0<\mu \leq 1,

here E_α, β(z) is the generalized two-parameter Mittag-Leffler function

$$E_{\alpha ,\beta }(z)=\sum\limits_{m=0}^\infty \frac{z^m}{\Gamma (\alpha m+\beta )},\qquad E_{\alpha ,1}(z)=E_\alpha (z).$$

Waiting time probability distribution function ψ_μ(τ) has the following asymptotic behavior (see, Ref.[1])

ψ_μ(τ) ≃ 1/ντ^μ + 1,   τ → ∞,

and

ψ_μ(τ) ≃ ντ^μ − 1,   τ → 0.

Fractional compound Poisson process

Fractional compound Poisson process has been introduced and developed by Nick Laskin (see, Ref.[1]). The fractional compound Poisson process {X(t), t ≥ 0} is represented by

$$X(t)=\sum\limits_{i=1}^{N(t)}Y_i,$$

where {N(t), t ≥ 0} is a fractional Poisson process, and {Y_i, i = 1, 2, …} is a family of independent and identically distributed random variables with probability distribution function p(Y) for each Y_i. The process {N(t), t ≥ 0} and the sequence {Y_i, i = 1, 2, …} are assumed to be independent.

The fractional compound Poisson process is natural generalization of the compound Poisson process.

See also

• Poisson process
• Poisson distribution
• Compound Poisson process
• Markov process
• Fractional calculus
• Generating function

Further reading

• N. Laskin, (2009), Some applications of the fractional Poisson probability distribution, J. Math. Phys. 50, 113513 (2009);(12 pages).
• L. Beghin and E. Orsingher, (2009), Fractional Poisson Processes and Related Planar Random Motions, Electronic Journal of Probability, Vol. 14 (2009), Paper no. 61, pages 1790–1826.
• M.M. Meerschaert, E. Nane, P. Vellaisamy, (2011), The Fractional Poisson Process and the Inverse Stable Subordinator, Electronic Journal of Probability, Vol. 16 (2011), Paper no. 59, pages 1600–1620.