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 {displaystyle u } , which has physical dimension [ ν ] = sec − μ {displaystyle =sec ^{-mu }} , where 0 < μ ≤ 1 {displaystyle 0<mu leq 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. 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 {displaystyle u } , which has physical dimension [ ν ] = sec − μ {displaystyle =sec ^{-mu }} , where 0 < μ ≤ 1 {displaystyle 0<mu leq 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. 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 functioncan 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. The probability distribution function of the fractional Poisson process has been found for the first time by Nick Laskin (see, Ref.) where parameter ν {displaystyle u } has physical dimension [ ν ] = sec − μ {displaystyle =sec ^{-mu }} and Γ ( μ ( k + n ) + 1 ) {displaystyle {Gamma (mu (k+n)+1)}} is the Gamma function. The P μ ( n , t ) {displaystyle P_{mu }(n,t)} gives us the probability that in the time interval [ 0 , t ] {displaystyle } we observe n events governed by fractional Poisson stream. The probability distribution of the fractional Poissonprocess P μ ( n , t ) {displaystyle P_{mu }(n,t)} can be represented in terms of the Mittag-Leffler function E μ ( z ) {displaystyle E_{mu }(z)} in the following compact way (see, Ref.),