Alternative Forms of Compound Fractional Poisson Processes

and Applied Analysis 3 where the first term refers to the probability mass concentrated in the origin, δ y denotes the Dirac delta function, and fYβ denotes the density of the absolutely continuous component. The function gYβ given in 1.5 satisfies the following fractional master equation, that is, ∂ ∂tβ gYβ ( y, t ) −λgYβ ( y, t ) λ ∫ ∞ −∞ gYβ ( y − x, t ) fX x dx, 1.6 where ∂/∂t is the Caputo fractional derivative of order β ∈ 0, 1 see, for example, 12 and the random variables Xj , j 1, 2, . . . have continuous density fX . We also recall the following result proved in 13 for the rescaled version of the timefractional compound Poisson process hereafter TFCPP : if the random variables Xj, j 1, 2, . . . are centered and have finite variance, then cYβ ct ⇒ W ( Lβ t ) , c −→ ∞, 1.7 where W is a standard Brownian motion and⇒ denotes weak convergence. A detailed exposition of the theory of TFCPP and continuous time random walks can be found in 14, 15 , where the density fYβ is expressed in terms of successive derivatives of the Mittag-Leffler function as follows: