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
Contents
- Fundamentals
- Fractional Poisson probability distribution function
- Mean
- The second order moment
- Variance
- Characteristic function
- Generating function
- Moment generating function
- Waiting time distribution function
- Fractional compound Poisson process
- Applications of fractional Poisson probability distribution
- Physical application New coherent states
- Mathematical applications New polynomials and numbers
- Statistical application and inference
- References
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.
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.
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 has been found for the first time by Nick Laskin (see, Ref.[1])
where parameter
The
The probability distribution of the fractional Poisson process
It follows from the above equations that when
where
Thus,
Mean
The mean
The second order moment
The second order moment of the fractional Poisson process
Variance
The variance of the fractional Poisson process is (see, Ref.[1])
where
Characteristic function
The characteristic function of the fractional Poisson process has been found for the first time in Ref.[1],
or in a series form
with the help of the Mittag-Leffler function series representation.
Then, for the moment of
Generating function
The generating function
The generating function of the fractional Poisson probability distribution was obtained for the first time by Nick Laskin in Ref.[1].
where
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
For example, for the moment of
The moment generating function
or in a series form
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
where
and
The waiting time probability distribution function
here
Waiting time probability distribution function
and
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
where
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
where
and
Then the probability
which is recognized as fractional Poisson probability distribution.
In terms of photon field creation and annihilation operators
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]).