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 that exhibits non-exponential distribution of interarrival times. The fractional Poisson process is a continuous-time process that 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, Fractional compound Poisson process and fractional Poisson probability distribution function have been invented, developed and encouraged for applications by Nick Laskin (2003) who coined the terms fractional Poisson process, Fractional compound Poisson process and fractional Poisson probability distribution function.[1]

## Contents

* 1 Fundamentals
* 2 Fractional Poisson probability distribution function
* 3 Mean
* 4 The second order moment
* 5 Variance
* 6 Characteristic function
* 7 Generating function
* 8 Moment generating function
* 9 Waiting time distribution function
* 10 Fractional compound Poisson process
* 11 Applications of fractional Poisson probability distribution
* 12 Physical application: New coherent states
* 13 Mathematical applications: New polynomials and numbers
* 14 Statistical application and inference
* 15 See also
* 16 References
* 17 Further reading

## 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 function 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.[2][3]

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 the fractional Poisson process has been found for the first time by Nick Laskin (see, Ref.[1])

Pμ(n,t)=(νtμ)nn!∑k=0∞(k+n)!k!(−νtμ)kΓ(μ(k+n)+1),0<μ≤1,

where parameter ν has physical dimension [ν]=sec−μ and Γ(μ(k+n)+1) is the Gamma function.

The Pμ(n,t) gives us the probability that in the time interval [0,t] we observe n events governed by fractional Poisson stream.

The probability distribution of the fractional Poisson process Pμ(n,t) can be represented in terms of the Mittag-Leffler function Eμ(z) in the following compact way (see, Ref.[1]),

Pμ(n,t)=((−z)nn!dndznEμ(z))|z=−νtμ,

Pμ(n=0,t)=Eμ(−νtμ).

It follows from the above equations that when μ=1 the Pμ(n,t) is transformed into the well known probability distribution function of the Poisson process, P(n,t)=P1(n,t), P(n,t)=(ν‾t)nn!exp⁡(−ν‾t),

P(n=0,t)=exp⁡(−ν‾t),

where ν‾ is the rate of arrivals with physical dimension [ν‾]=sec−1.

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

## Mean

The mean n‾μ of the fractional Poisson process has been found in Ref.[1].

n‾μ=∑n=0∞nPμ(n,t)=νtμΓ(μ+1).

## The second order moment

The second order moment of the fractional Poisson process n2‾μ has been found for the first time by Nick Laskin (see, Ref.[1])

nμ2‾=∑n=0∞n2Pμ(n,t)=n‾μ+n‾μ2πΓ(μ+1)22μ−1Γ(μ+12).

## Variance

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

σμ=nμ2‾−n‾μ2=n‾μ+n‾μ2{μB(μ,12)22μ−1−1},

where B(μ,12) is the Beta-function.

## Characteristic function

The characteristic function of the fractional Poisson process has been found for the first time in Ref.[1],

Cμ(s,t)=∑n=0∞eisnPμ(n,t)=Eμ(νtμ(eis−1)).

or in a series form

Cμ(s,t)=∑m=0∞1Γ(mμ+1)(νtμ(eis−1))m,

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

Then, for the moment of kth order we have

nμk‾=(1/ik)∂kCμ(s,t)∂sk|s=0.

## Generating function

The generating function Gμ(s,t) of the fractional Poisson probability distribution function is defined as (see, Ref.[1]).

Gμ(s,t)=∑n=0∞snPμ(n,t).

The generating function of the fractional Poisson probability distribution was obtained for the first time by Nick Laskin in Ref.[1].

Gμ(s,t)=Eμ(νtμ(s−1)),

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

Eμ(z)=∑m=0∞zmΓ(μm+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 Hμ(s,t) which is defined as

Hμ(s,t)=∑n=0∞e−snPμ(n,t).

For example, for the moment of kth order we have

nμk‾=(−1)k∂kHμ(s,t)∂sk|s=0.

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

Hμ(s,t)=Eμ(νtμ(e−s−1)),

or in a series form

Hμ(s,t)=∑m=0∞1Γ(mμ+1)(νtμ(e−s−1))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 ψμ(τ) of the fractional Poisson process is defined as (see, Refs.[1,3])

ψμ(τ)=−ddτPμ(τ),

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

Pμ(τ)=1−∑n=1∞Pμ(n,τ)=Eμ(−ντμ),

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

The waiting time probability distribution function ψμ(τ) of the fractional Poisson process was found for first time by Nick Laskin in Ref.[1],

ψμ(τ)=ντμ−1Eμ,μ(−ντμ),t≥0,0<μ≤1,

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

Eα,β(z)=∑m=0∞zmΓ(αm+β),Eα,1(z)=Eα(z).

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

ψμ(τ)≃1/ντμ+1,τ->∞,

and

ψμ(τ)≃ντμ−1,τ->0.

## Fractional compound Poisson process

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

X(t)=∑i=1N(t)Yi,

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

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

## Applications of fractional Poisson probability distribution

The fractional Poisson probability distribution has physical and mathematical applications. Physical application is in the field of quantum optics. Mathematical applications are in the field of combinatorial numbers (see, Ref.[4]).

## Physical application: New coherent states

A new family of quantum coherent states |ς> has been introduced as[4]

|ς⟩=∑n=0∞(μςμ)nn!(Eμ(n)(−μ|ς|2μ))1/2|n⟩,

where |n⟩ is an eigenvector of the photon number operator, complex number ς stands for labeling the new coherent states,

Eμ(n)(−μ|ς|2μ)=dndznEμ(z)|z=−μ|ς|2μ

and Eμ(x) is the Mittag-Leffler function.

Then the probability Pμ(n) of detecting n photons is:

Pμ(n)=|⟨n∣ς⟩|2=(μ|ς|2μ)nn!(Eμ(n)(−μ|ς|2μ)),

which is recognized as fractional Poisson probability distribution.

In terms of photon field creation and annihilation operators a+ and a that satisfy the canonical commutation relation [a,a+]=aa+−a+a=1, the average number of photons n¯ in a coherent state |ς⟩ can be presented as (see, Ref.[4])

n¯=⟨ς|a+a|ς⟩=∑n=0∞nPμ(n)=(μ|ς|2μ)/Γ(μ+1).

## Mathematical applications: New polynomials and numbers

The fractional generalization of Bell polynomials, Bell numbers, Dobinski's formula and Stirling numbers of the second kind have been introduced and developed by Nick Laskin (see, Ref.[4]). The appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been applied to evaluate the skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been discovered (see, Ref.[4]).

In the limit case μ = 1 when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed applications turn into the well-known results of the quantum optics and the enumerative combinatorics.

## Statistical application and inference

The point and interval estimators for the model parameters are developed by Cahoy et. al, (2010) (see, Ref.[5]). [5]

## See also

* Poisson process
* Poisson distribution
* Compound Poisson process
* Markov process
* Fractional calculus
* Generating function
* Coherent states
* Canonical commutation relation
* Bell polynomials
* Bell numbers
* Dobiński's formula
* Stirling numbers
* Mittag-Leffler distribution

## References

1. ↑ Laskin, N. (2003). "Fractional Poisson process". Communications in Nonlinear Science and Numerical Simulation 8 (3–4): 201–213. doi:10.1016/S1007-5704(03)00037-6.
2. ↑ Saichev, A.I.; Zaslavsky, G.M. (1997). "Fractional kinetic equations: solutions and applications". Chaos 7 (4): 753–764. doi:10.1063/1.166272. PMID 12779700.
3. ↑ O. N. Repin and A. I. Saichev, (2000), Fractional Poisson Law, Radiophysics and Quantum Electronics, vol 43, Number 9 (2000), 738-741, https://doi.org/10.1023/A:1004890226863.
4. ↑ N. Laskin, (2009), Some applications of the fractional Poisson probability distribution, J. Math. Phys. 50, 113513 (2009) (12 pages), http://jmp.aip.org/resource/1/jmapaq/v50/i11/p113513_s1?bypassSSO=1[yes|permanent dead link|dead link}}]. (also available online: https://arxiv.org/abs/0812.1193)
5. ↑ D.O. Cahoy V.V. Uchaikin W.A. Woyczyński (2010). "Parameter estimation for fractional Poisson processes". Journal of Statistical Planning and Inference 140 (11): 3106–3120. doi:10.1016/j.jspi.2010.04.016.

## Further reading

* 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.

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