Alternative Forms of Compound Fractional Poisson Processes

Abstract

We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one. Only in the final case treated in this paper, where the number of jumps is given by the fractional-difference Poisson process defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with order greater than one. Moreover, in this case, the compound Poisson process is Markovian and this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random walks with stable processes (or inverse stable subordinators). These subordinating relationships hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy master equations which are the fractional analogues of the well-known Kolmogorov one.

1. Introduction and Preliminary Results

The fractional Poisson process (FPP), which we will denote by 𝒩_𝛽(𝑡), 𝑡>0, 𝛽∈(0,1], has been introduced in [1], by replacing, in the differential equation governing the Poisson process, the time derivative with a fractional one. Later, in [2, 3], it was proved to be a renewal process with Mittag-Leffler distributed waiting times (and therefore with infinite mean). In [4] it has been expressed as the composition 𝑁(𝒯_𝛽(𝑡)) of a standard Poisson process 𝑁 with the fractional diffusion 𝒯_𝛽, independent of 𝑁. A full characterization of 𝒩_𝛽 in terms of its finite multidimensional distributions can be found in [5]. In [6] the coincidence between 𝒩_𝛽 and the fractal time Poisson process (FTPP) defined as 𝑁(ℒ_𝛽(𝑡)) has been proved, where ℒ_𝛽(𝑡), 𝑡≥0 is the inverse of the stable subordinator 𝒜_𝛽(𝑡) of index 𝛽 (with parameters 𝜇=0, 𝜃=1,  𝜎=(𝑡cos𝜋𝛽/2)^1/𝛽, in the notation of [7], that we will adopt hereafter). Thus, the process 𝒜_𝛽 is characterized by the following Laplace pairs: 𝔼𝑒^{−𝑘𝒜𝛽(𝑡)}=𝑒^{−𝑘𝛽𝑡},𝑘,𝑡>0,_0+∞𝑒^−𝑠𝑡ℎ_𝛽(𝑥,𝑡)𝑑𝑡=𝑥^𝛽−1𝐸_𝛽,𝛽−𝑠𝑥^𝛽,𝑠,𝑥>0, (1.1) where 𝐸_𝛽,𝛿 is the Mittag-Leffler function of parameters 𝛽, 𝛿 and ℎ_𝛽(𝑥,𝑡) is the density of 𝒜_𝛽(𝑡). The inverse stable subordinator ℒ_𝛽 is defined by the following relation: ℒ_𝛽(𝑡)∶=inf𝑠∶𝒜_𝛽(𝑠)=𝑡,𝑧,𝑡>0, (1.2) and therefore we get 𝔼𝑒^{−𝑘ℒ𝛽(𝑡)}=𝐸_𝛽,1−𝑘𝑡^𝛽,𝑘,𝑡>0,_0+∞𝑒^−𝑠𝑡𝑙_𝛽(𝑥,𝑡)𝑑𝑡=𝑠^𝛽−1𝑒^{−𝑥𝑠𝛽},𝑠,𝑥>0, (1.3) where 𝑙_𝛽(𝑥,𝑡) is the density of ℒ_𝛽(𝑡).

We will make use also of different forms of FPP such as the alternative fractional Poisson process in [8] and the fractional-difference Poisson process in [9].

In this paper we study several fractional compound Poisson processes and, to help the reader, we list the acronyms used throughout the paper by the end of the paper.

The first form of fractional compound Poisson process has been introduced in [10], in the form of a continuous time random walk with infinite-mean waiting times (see also [11]). This corresponds to the following random walk time changed via the FTPP, that is, 𝑌_𝛽(𝑡)=^{𝑁(ℒ𝛽(𝑡))}_𝑗=1𝑋_𝑗,𝑡≥0, (1.4) with 𝑋_𝑗,𝑗=1,2,…  are i.i.d. random variables, independent from 𝑁 and ℒ_𝛽. The last assumption (that we will adopt throughout the paper) corresponds to the so-called uncoupled case.

In [6] it is proved that subordinating random walk to the fractional Poisson process 𝒩_𝛽(𝑡), 𝑡≥0, produces the same one-dimensional distribution. The (generalized) density function of 𝑌_𝛽(𝑡) can be expressed as 𝑔_𝑌𝛽𝑦,𝑡∶=𝐸_𝛽,1−𝜆𝑡^𝛽𝛿𝑦+𝑓_𝑌𝛽𝑦,𝑡,𝑦,𝑡≥0, (1.5) where the first term refers to the probability mass concentrated in the origin, 𝛿(𝑦) denotes the Dirac delta function, and 𝑓_𝑌𝛽 denotes the density of the absolutely continuous component. The function 𝑔_𝑌𝛽 given in (1.5) satisfies the following fractional master equation, that is, 𝜕^𝛽𝜕𝑡^𝛽𝑔_𝑌𝛽𝑦,𝑡=−𝜆𝑔_𝑌𝛽𝑦,𝑡+𝜆^+∞_−∞𝑔_𝑌𝛽𝑦−𝑥,𝑡𝑓_𝑋(𝑥)𝑑𝑥, (1.6) where 𝜕^𝛽/𝜕𝑡^𝛽 is the Caputo fractional derivative of order 𝛽∈(0,1] (see, for example, [12]) and the random variables 𝑋_𝑗, 𝑗=1,2,… have continuous density 𝑓_𝑋.

We also recall the following result proved in [13] for the rescaled version of the time-fractional compound Poisson process (hereafter TFCPP): if the random variables 𝑋_𝑗, 𝑗=1,2,… are centered and have finite variance, then 𝑐^−𝛽/2𝑌_𝛽(𝑐𝑡)⇒𝑊ℒ_𝛽(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 𝑓_𝑌𝛽 is expressed in terms of successive derivatives of the Mittag-Leffler function as follows: 𝑓_𝑌𝛽𝑦,𝑡=^∞_𝑛=1𝑓_𝑋^∗𝑛𝑦Pr𝒩_𝛽(𝑡)=𝑛=^∞_𝑛=1𝑓_𝑋^∗𝑛𝑦𝜆𝑡^𝛽𝑛𝜕𝑛!𝜕𝑥^𝑛𝑛𝐸_𝛽,1(𝑥)_{𝑥=−𝜆𝑡𝛽},𝑡,𝑦≥0, (1.8) where 𝑓_𝑋^∗𝑛 is the 𝑛th convolution of the density 𝑓_𝑋 of the r.v.'s 𝑋_𝑗.

A further asymptotic result has been proved in [15], under the assumption that the density of the jump variables (which we will denote, in this special case, as 𝑋^∗_𝑗) behaves asymptotically as 𝑓_𝑋∗(ℎ𝜅)∶=^+∞_−∞𝑒^{𝑖𝜅ℎ𝑥}𝑓_𝑋∗(𝑥)𝑑𝑥≃1−ℎ^𝛼|𝜅|^𝛼,ℎ→0,𝛼∈(0,1], (1.9) where ⋅̂ denotes the Fourier transform. In this case the TFCPP is defined as 𝑌^∗_𝛽(𝑡)=∑^{𝒩𝛽(𝑡)}_𝑗=1𝑋^∗_𝑗 and the rescaled version displays the following weak convergence: ℎ𝑌^∗_𝛽𝑡𝑟⇒𝑍(𝑡), (1.10) for ℎ,𝑟→0, s.t. ℎ^𝛼/𝑟^𝛽→1. The characteristic function of the limiting process 𝑍(𝑡) is given by 𝐸_𝛽,1−𝜆𝑡^𝛽|𝜅|^𝛼 (1.11) and thus it can be represented as 𝒮_𝛼(ℒ_𝛽(𝑡)), where 𝒮_𝛼 is a symmetric 𝛼-stable process with parameters 𝜇=0, 𝜃=0, 𝜎=(𝑡cos𝜋𝛼/2)^1/𝛼. For 𝛽<1, the inverse stable subordinator ℒ_𝛽(𝑡) is not Markovian as well as not Lévy (see [10]) and the same is true for 𝒮_𝛼(ℒ_𝛽(𝑡)), as remarked in [15]; moreover, the density 𝑢=𝑢(𝑦,𝑡) of the latter is the solution to the space-time fractional equation: 𝜕^𝛽𝑢𝜕𝑡^𝛽𝜕=𝜆^𝛼𝑢𝜕𝑦^𝛼,𝑢𝑦,0=𝛿𝑦,𝑦∈ℝ,𝑡>0, (1.12) where 𝜕^𝛼/𝜕|𝑦|^𝛼 denotes the Riesz-Feller derivative of order 𝛼∈(0,1] (see [16]). Thus, in the special case 𝛼=1, it reduces to the composition of a Cauchy process with ℒ_𝛽.

Finally, we recall the following result proved in [17]: under the assumption of heavy tailed r.v.’s representing the jumps, that is, Pr𝑋_𝑗>𝑥∼𝑥^−𝛼,𝑥→∞, (1.13) the following convergence holds, as 𝑐→∞, 𝑐^−𝛽/𝛼𝑌_𝛽(𝑐𝑡)⇒ℐℒ_𝛽(𝑡). (1.14) In (1.14) ℐ is a 𝛼-stable Lévy process with density 𝑝_𝛼(𝑥,𝑡) and characteristic function 𝑝_𝛼(𝜅,𝑡)=𝑒^{𝑡𝑏[𝑞(−𝑖𝜅)𝛼+(1−𝑞)(𝑖𝜅)𝛼]},for𝑏<0,0<𝛼<1or𝑏>0,1<𝛼<2, (1.15) under the assumption that lim_𝑥→∞Pr{𝑋_𝑗<−𝑥}/Pr{|𝑋_𝑗|>𝑥}=𝑞∈0,1]. The density of the limiting process is proved to satisfy the following time and space fractional equation: 𝐷^𝛽_0+,𝑡𝑢=𝑞𝑏𝐷^𝛼_−,𝑥𝑢+1−𝑞𝑏𝐷^𝛼_0+,𝑥𝑢, (1.16) where the fractional derivatives are intended in the Riemann-Liouville sense (see [12], formulae (2.2.3) and (2.2.4), page 80).

We present, in this paper, different versions of the compound Poisson process (CPP), fractional (under different acceptions) with respect to time and space; we provide for them analytic expressions of the distributions and some composition relationships with stable and inverse-stable processes, holding not only in the scaling limit, but also in the finite domain.

Tables 1 and 2 provide a summary of these results in the finite and asymptotic domains, respectively.

Table 1 

Main results in finite domain.

Table 2 

Main results in asymptotic domain.

We assume here exponential jumps (generalized later to Mittag-Leffler), since this allows to obtain explicit equations (fractional in most cases) driving these fractional CPP's for any finite value of the time and space arguments. This kind of explicit formulae, together with the knowledge of the related governing differential equations, is of great importance in many actuarial applications (see, for example, [18], Section 4.2). In risk theory it is related to the Tweedie's compound Poisson model (see [19]). The hypothesis of exponential jumps has been widely applied also in other fields: in natural sciences it leads to the so-called compound Poisson-Gamma model, which is used for rainfall prediction (see, for example, [20]).

2. Time-Fractional Compound Poisson Processes

We consider different forms of TFCPP, starting with the more familiar one given in (1.4) and then comparing the results with those obtained for an alternative definition of FPP.

2.1. The Standard Case

In order to get a form of the density of the TFCPP more explicit than (1.8), we assume that the 𝑋_𝑗's are exponentially distributed: in this case it can be expressed in terms of the generalized Mittag-Leffler function: 𝐸^𝛾_𝛼,𝛿(𝑥)=^∞_𝑗=0𝛾^(𝑗)𝑥𝑗!Γ^𝑗(𝛼)(𝛿)𝛼𝑗+𝛿,𝛼,𝛿,𝛾∈ℂ,ℛ,ℛ>0, (2.1) where (𝑥)^(𝑛)=𝑥(𝑥+1)⋯(𝑥+𝑛−1) is the rising factorial (or Pochhammer symbol). Moreover, we can obtain the fractional partial-differential equation satisfied by the density of its absolutely continuous component.

Theorem 2.1. The process 𝑌_𝛽(𝑡)=^{𝒩𝛽(𝑡)}_𝑗=1𝑋_𝑗,𝑡≥0, (2.2) with 𝑋_𝑗, 𝑗=1,2…, independent and exponentially distributed with parameter 𝜉, has the following distribution: Pr𝑌_𝛽(𝑡)≤𝑦=𝐸_𝛽,1−𝜆𝑡^𝛽1_[0,+∞)𝑦+^𝑦_−∞𝑓_𝑌𝛽(𝑧,𝑡)𝑑𝑧,𝑡≥0,𝑦∈ℝ, (2.3) where 𝑓_𝑌𝛽=𝑒𝑦,𝑡^−𝜉𝑦𝑦^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛()!𝐸𝑛−1^{𝑛+1𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽1_[0,+∞)𝑦,𝑡≥0. (2.4) The function 𝑓_𝑌𝛽(𝑦,𝑡) given in (2.4) satisfies the following partial differential equation: 𝜉𝜕^𝛽𝜕𝑡^𝛽𝑓_𝑌𝛽𝜕=−𝜆+^𝛽𝜕𝑡^𝛽𝜕𝑓𝜕𝑦^𝑌𝛽,𝑡,𝑦≥0, (2.5) where 𝜕^𝛽/𝜕𝑡^𝛽 denotes the Caputo fractional derivative with the conditions 𝑓_𝑌𝛽𝑦,0=0,_0+∞𝑓_𝑌𝛽𝑦,𝑡𝑑𝑦=1−𝐸_𝛽,1−𝜆𝑡^𝛽. (2.6)

Proof. Formula (1.8) can be rewritten by considering that 𝑓_𝑋^∗𝑛(𝑦)=𝜉^𝑛𝑦^𝑛−1𝑒^−𝜉𝑦/(𝑛−1)! and using the expression of Pr{𝒩_𝛽(𝑡)=𝑛} in terms of generalized Mittag-Leffler functions (see [21]), that is, Pr𝒩_𝛽(𝑡)=𝑛=𝜆^𝑛𝑡^𝑛𝛽𝐸^𝑛+1_{𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽,𝑛≥0. (2.7) In order to derive (2.5), we evaluate the following partial derivatives of (2.4): 𝜕^𝛽𝜕𝑡^𝛽𝑓_𝑌𝛽=𝑒^−𝜉𝑦𝑦𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛(𝑛−1)!𝑛!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗,𝜕𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1𝑓𝜕𝑦^𝑌𝛽𝑦,𝑡=−𝜉𝑒^−𝜉𝑦𝑦^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛(𝑛−1)!𝑛!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗++𝑒𝑗!Γ𝛽𝑗+𝛽𝑛+1_−𝜉𝑦𝑦^2∞_𝑛=2𝜆𝜉𝑡^𝛽𝑦^𝑛(𝑛−2)!𝑛!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗,𝜕𝑗!Γ𝛽𝑗+𝛽𝑛+1𝜕𝜕𝑦𝜕𝑡^𝛽𝛽𝑓_𝑌𝛽𝑦,𝑡=−𝜉𝑒^−𝜉𝑦𝑦𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛𝑛!(𝑛−1)!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗++𝑒𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1_−𝜉𝑦𝑦²𝑡^𝛽∞_𝑛=2𝜆𝜉𝑡^𝛽𝑦^𝑛(𝑛−2)!𝑛!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1=−𝜉𝑒^−𝜉𝑦𝑦𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛((𝑛−1)!)^2∞_𝑗=0𝑛+𝑗−1!−𝜆𝑡^𝛽𝑗+−𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1𝜉𝑒^−𝜉𝑦𝑦𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛𝑛!(𝑛−1)!^∞_𝑗=1𝑛+𝑗−1!−𝜆𝑡^𝛽𝑗++𝑒𝑗−1!Γ𝛽𝑗+𝛽𝑛−𝛽+1_−𝜉𝑦𝑦²𝑡^𝛽∞_𝑛=2𝜆𝜉𝑡^𝛽𝑦^𝑛(𝑛−2)!(𝑛−1)!^∞_𝑗=0𝑛+𝑗−1!−𝜆𝑡^𝛽𝑗++𝑒𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1_−𝜉𝑦𝑦²𝑡^𝛽∞_𝑛=2𝜆𝜉𝑡^𝛽𝑦^𝑛(𝑛−2)!𝑛!^∞_𝑗=0𝑛+𝑗−1!−𝜆𝑡^𝛽𝑗.𝑗−1!Γ𝛽𝑗+𝛽𝑛−𝛽+1 (2.8) By inserting (2.8) in (2.5), the equation is satisfied. Finally, it can be easily verified that the initial condition holds. In order to check the second condition in (2.6), we integrate 𝑓_𝑌𝛽 with respect to 𝑦: ^∞₀𝑒^−𝜉𝑦𝑦^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝑛()!𝐸𝑛−1^{𝑛+1𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽𝑑𝑦=^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑛()!()!𝑛−1𝑛−1𝜉^𝑛𝐸^𝑛+1_{𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽=^∞_𝑛=0𝜆𝑡^𝛽𝑛𝐸^𝑛+1_{𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽−𝐸_𝛽,1−𝜆𝑡^𝛽=1−𝐸_𝛽,1−𝜆𝑡^𝛽, (2.9) where, in the last step, we have applied formula (2.30) of [21], for 𝑢=1.

2.1.1. The Nonfractional Case 𝛽=1

From (2.4), we obtain the distribution of the standard CPP, defined as 𝑌(𝑡)=∑^𝑁(𝑡)_𝑛=1𝑋_𝑗, under the assumption of exponential jumps 𝑋_𝑗, which reads Pr𝑌(𝑡)≤𝑦=𝑒^−𝜆𝑡1_[0,+∞)𝑦+^𝑦_−∞𝑓_𝑌(𝑧,𝑡)𝑑𝑧,𝑡≥0,𝑦∈ℝ, (2.10) where 𝑓_𝑌=𝑒𝑦,𝑡^{−𝜉𝑦−𝜆𝑡}𝑦^∞_𝑛=1𝜆𝜉𝑡𝑦^𝑛()!1𝑛!𝑛−1^[0,+∞)𝑦=𝜆𝜉𝑡𝑒^{−𝜉𝑦−𝜆𝑡}𝑊_1,2𝜆𝜉𝑡𝑦1_[0,+∞)𝑦𝑊,𝑡≥0,_𝛼,𝛽(𝑧)=^∞_𝑗=0𝑧^𝑗𝑗!Γ𝛼𝑗+𝛽,𝛼>−1,𝛽,𝑧∈ℂ, (2.11)(2.12) is the Wright function. Equation (4.2.8) in [18] provides another expression of 𝑓_𝑌 in terms of the modified Bessel function. The density (2.11) satisfies the following equation: 𝜉𝜕𝑓𝜕𝑡^𝑌𝜕=−𝜆+𝜕𝜕𝑡𝑓𝜕𝑦^𝑌, (2.13) with conditions 𝑓_𝑌𝑦,0=0,_0+∞𝑓_𝑌𝑦,𝑡𝑑𝑦=1−𝑒^−𝜆𝑡, (2.14) as can be easily verified directly.

Now we recall the following subordination law presented in [6] in a more general setting: 𝑌_𝛽(𝑡)=^𝑑𝑌ℒ_𝛽(𝑡), (2.15) where ℒ_𝛽(𝑡), 𝑡≥0 is the inverse stable subordinator defined by (1.2). We give an explicit proof of (2.15), which will be useful to prove analogous results in the next sections. We start with the evaluation of the Laplace transform (hereafter denoted by  ⋅̃) of 𝑌_𝛽(𝑡) with respect to 𝑦: by considering the probability generating function of 𝒩_𝛽, that is, 𝔼𝑢^{𝒩𝛽(𝑡)}=𝐸_𝛽,1−𝜆𝑡^𝛽(1−𝑢),|𝑢|≤1, (2.16) we get 𝑔_𝑌𝛽(𝑘,𝑡)∶=𝔼𝑒^{−𝑘𝑌𝛽(𝑡)}=𝐸_𝛽,1−𝜆𝑘𝑡𝑘+𝜉^𝛽. (2.17) Formula (2.17), Laplace transformed with respect to 𝑡, gives 𝑔_𝑌𝛽(𝑘,𝑠)∶=₀^+∞𝑒^−𝑠𝑡𝑔_𝑌𝛽()𝑠𝑘,𝑡𝑑𝑡=^𝛽−1(𝑘+𝜉)𝑠^𝛽(),𝑘+𝜉+𝑘𝜆 (2.18) which can be rewritten as 𝑔_𝑌𝛽(𝑘,𝑠)=𝑠^𝛽−1₀^+∞𝑒^{−𝑠𝛽𝑡}𝔼𝑒^{−𝑘𝑌(𝑡)}=(=𝑑𝑡by1.3)_0+∞𝔼𝑒^{−𝑘𝑌(𝑧)}𝑙_𝛽̃(𝑧;𝑠)𝑑𝑧, (2.19) where 𝑙_𝛽̃(𝑧;𝑠)∶=∫₀^+∞𝑒^−𝑠𝑡𝑙_𝛽(𝑧,𝑡)𝑑𝑡. Thus, by inverting the double Laplace transform, we get Pr𝑌_𝛽(𝑡)∈𝑑𝑦=₀^+∞Pr𝑌(𝑧)∈𝑑𝑦𝑙_𝛽(𝑧,𝑡)𝑑𝑧. (2.20) Now it is also easy to derive (2.5), since we can write in particular from (2.20) that 𝑓_𝑌𝛽𝑦,𝑡=₀^+∞𝑓_𝑌𝑦,𝑧𝑙_𝛽(𝑧,𝑡)𝑑𝑧 (2.21) and thus we get 𝜕^𝛽𝜕𝑡^𝛽𝑓_𝑌𝛽𝑦,𝑡=₀^+∞𝑓_𝑌𝜕𝑦,𝑧^𝛽𝜕𝑡^𝛽𝑙_𝛽(𝑧,𝑡)𝑑𝑧=−₀^+∞𝑓_𝑌𝜕𝑦,𝑧𝑙𝜕𝑧^𝛽(𝑧,𝑡)𝑑𝑧. (2.22) Indeed, it is well known that ℒ_𝛽(𝑡) is governed by the following equation: 𝜕^𝛽𝜕𝑡^𝛽𝑙_𝛽()𝜕𝑧,𝑡=−𝑙𝜕𝑧^𝛽(𝑧,𝑡),𝑙_𝛽(𝑧,0)=𝛿(𝑧),𝑧,𝑡≥0. (2.23) By integrating by parts and applying the initial condition, (2.22) becomes 𝜕^𝛽𝜕𝑡^𝛽𝑓_𝑌𝛽𝑦,𝑡=₀^+∞𝜕𝑓𝜕𝑧^𝑌𝑦,𝑧𝑙_𝛽()=(1𝑧,𝑡𝑑𝑧by2.13)=−𝜉𝜕𝜕𝑦₀^+∞𝜕𝑓𝜕𝑧^𝑌𝑦,𝑧𝑙_𝛽()𝜆𝑧,𝑡𝑑𝑧−𝜉𝜕𝜕𝑦₀^+∞𝑓_𝑌𝑦,𝑧𝑙_𝛽()1𝑧,𝑡𝑑𝑧=−𝜉𝜕𝜕𝜕𝑦𝜕𝑡^𝛽𝛽𝑓_𝑌𝛽−𝜆𝑦,𝑡𝜉𝜕𝑓𝜕𝑦^𝑌𝛽𝑦,𝑡. (2.24)

2.2. An Alternative Case

We consider now a different model of TFCPP, based on the alternative definition of FPP given in [4], that is, Pr𝑁_𝛽(𝑡)=𝑘=𝜆𝑡^𝛽𝑘Γ1𝛽𝑘+1𝐸_𝛽,1𝜆𝑡^𝛽,𝑡,𝑘≥0. (2.25) The process with the above state probabilities plays a crucial role in the evolution of some random motions (see [22]) and can be considered as a fractional version of the Poisson process because its probability generating function (displayed below) satisfies a fractional equation (see formula (4.5) of [4]). The distribution (2.25) can be interpreted as a weighted Poisson distribution (for the general concept of discrete weighted distribution see, e.g., [23], page 90, and the references cited therein) and, as explained in [8], the weights that do not depend on 𝑡; actually we have Pr𝑁_𝛽(𝑡)=𝑤=𝑘^𝑘𝑝_𝑘𝑡^𝛽∑_𝑗≥0𝑤_𝑗𝑝_𝑗𝑡^𝛽,𝑡,𝑘≥0, (2.26) where 𝑤_𝑗=𝑗!/Γ(𝛽𝑗+1),  𝑗=0,1,… (for all 𝑡) and 𝑝_𝑗(𝑡)=((𝜆𝑡)^𝑗/𝑗!)𝑒^−𝜆𝑡, 𝑗=0,1,… are the distribution of the standard Poisson process 𝑁 with intensity 𝜆. We also recall [24] where one can find a sample path version of the weighted Poisson process.

We remark that the corresponding process is not Markovian, as 𝒩_𝛽, and moreover is not a renewal. Nevertheless, it is, for some aspects, more similar to the standard Poisson process 𝑁 than 𝒩_𝛽. For example, the rate of the asymptotic behavior of its moments is the same as for 𝑁.

The moment generating function is given by 𝔼𝑒^{𝜃𝑁𝛽(𝑡)}=𝐸_𝛽,1𝜆𝑡^𝛽𝑒^𝜃𝐸_𝛽,1𝜆𝑡^𝛽, (2.27) so that we get 𝔼𝑁_𝛽(𝑡)=𝜆𝑡^𝛽𝛽𝐸_𝛽,𝛽𝜆𝑡^𝛽𝐸_𝛽,1𝜆𝑡^𝛽. (2.28) By applying the following asymptotic formula of the Mittag-Leffler function 𝐸_𝛽,𝜈(𝑧)≃1𝛽𝑧^{(1−𝜈)/𝛽}exp𝑧^1/𝛽,as𝑧→∞, (2.29) (see, for example, [25] or [26]) we get 𝔼𝑁_𝛽(𝑡)≃1𝛽𝜆^1/𝛽𝑡,as𝑡→∞, (2.30) while for 𝒩_𝛽 the mean value behaves asymptotically as 𝑡^𝛽.

We define the alternative TFCPP as 𝑌_𝛽(𝑡)=^{𝑁𝛽(𝑡)}_𝑗=1𝑋_𝑗,𝑡≥0,𝛽∈(0,1], (2.31) where again 𝑋_𝑗’s are i.i.d. with exponential distribution, independent from 𝑁_𝛽. Under this assumption we obtain the following result on the distribution of 𝑌_𝛽.

Theorem 2.2. The process 𝑌_𝛽 defined in (2.31), with 𝑋_𝑗, 𝑗=1,2,…, independent and exponentially distributed with parameter 𝜉, has the following distribution: Pr𝑌_𝛽(𝑡)=1≤𝑦𝐸_𝛽,1𝜆𝑡^𝛽1_[0,+∞)𝑦+^𝑦_−∞𝑓_𝑌𝛽(𝑧,𝑡)𝑑𝑧,𝑡≥0,𝑦∈ℝ, (2.32) where 𝑓_𝑌𝛽𝑦,𝑡=𝜆𝜉𝑡^𝛽𝑒^−𝜉𝑦𝐸_𝛽,1𝜆𝑡^𝛽𝑊_𝛽,𝛽+1𝜆𝜉𝑡^𝛽𝑦1_[0,+∞)𝑦,𝑡≥0. (2.33)

Proof. The density (2.33) can be obtained as follows: 𝑓_𝑌𝛽=𝑒𝑦,𝑡^−𝜉𝑦𝐸_𝛽,1𝜆𝑡^𝛽∞_𝑛=1𝜆𝑡^𝛽𝑛Γ𝜉𝛽𝑛+1^𝑛𝑦^𝑛−1()!=𝑛−1𝜆𝜉𝑡^𝛽𝑒^−𝜉𝑦𝐸_𝛽,1𝜆𝑡^𝛽∞_𝑙=0𝜆𝜉𝑡^𝛽𝑙.𝑙!Γ𝛽𝑙+𝛽+1 (2.34) Moreover, one can check that ^∞₀𝑓_𝑌𝛽1𝑦,𝑡𝑑𝑦=1−𝐸_𝛽,1𝜆𝑡^𝛽, (2.35) and this completes the proof.

Remark 2.3. For 𝛽=1, formula (2.33) reduces to (2.11). We note that, as happens for the standard case, the density in (2.33) is expressed in terms of a single Wright function instead of an infinite sum of generalized Mittag-Leffler functions (as for the process 𝑌_𝛽). Nevertheless, the presence of a Mittag-Leffler in the denominator does not allow to evaluate the equation satisfied by 𝑓_𝑌𝛽.

2.2.1. Asymptotic Results

The analogy with the standard case is even more evident in the asymptotic behavior of the rescaled version of (2.31). Under the assumption (1.9) for the r.v.'s 𝑋^∗_𝑗, we can prove that, as ℎ,𝑟→0, s.t. ℎ^𝛼/𝑟→1 (not depending on 𝛽), ℎ𝑌^∗_𝛽𝑡𝑟=^{𝑁𝛽(𝑡/𝑟)}_𝑗=1ℎ𝑋^∗_𝑗⇒𝒮^𝛽_𝛼(𝑡), (2.36) where 𝒮^𝛽_𝛼 is a symmetric 𝛼-stable Lévy process with 𝜇=𝜃=0 and 𝜎=((1/𝛽)𝜆^1/𝛽𝑡cos(𝜋𝛼/2))^1/𝛼. Indeed, the characteristic function of (2.36) can be written as 𝑔_ℎ𝑌^∗_𝛽𝑡𝜅,𝑟=𝔼𝑒^{𝑖𝜅ℎ𝑌∗𝛽(𝑡/𝑟)}=1𝐸_𝛽,1𝜆𝑡^𝛽/𝑟^𝛽∞_𝑛=0𝜆𝑡^𝛽/𝑟^𝛽𝑓_ℎ𝑋(𝜅)^𝑛Γ=𝐸𝛽𝑛+1_𝛽,1𝜆𝑓_ℎ𝑋(𝜅)𝑡^𝛽/𝑟^𝛽𝐸_𝛽,1𝜆𝑡^𝛽/𝑟^𝛽=(≃𝐸bytheassumption1.9)_𝛽,1𝜆𝑡^𝛽/𝑟^𝛽−𝜆𝑡^𝛽ℎ^𝛼|𝜅|^𝛼/𝑟^𝛽𝐸_𝛽,1𝜆𝑡^𝛽/𝑟^𝛽=[(𝜆for2.29)]≃exp^1/𝛽𝑡𝑟1−ℎ^𝛼|𝜅|^𝛼1/𝛽−1. (2.37) By considering the generalized binomial theorem, we get from (2.37) that 𝑔_ℎ𝑌^∗_𝛽𝑡𝜅,𝑟𝜆≃exp^1/𝛽𝑡𝑟^∞_𝑗=01𝛽𝑗−ℎ^𝛼|𝜅|^𝛼𝑗𝜆=exp^1/𝛽𝑡𝑟ℎ1−^𝛼|𝜅|^𝛼𝛽+𝑜ℎ^𝛼. (2.38) Therefore, the limiting process is represented by the 𝛼-stable process 𝒮^𝛽_𝛼 with characteristic function 𝑒^{−(1/𝛽)𝜆1/𝛽𝑡|𝜅|𝛼}, instead of the subordinated process 𝒮_𝛼(ℒ_𝛽(𝑡)) obtained in the limit when considering the FPP 𝒩_𝛽; note that 𝒮_𝛼(ℒ_𝛽(𝑡)) coincides with 𝑍(𝑡) in (1.10). It is clear that the dependence on 𝛽 is limited to the scale parameter; the space-fractional equation satisfied by its density is therefore given by 𝜕𝑢=𝜆𝜕𝑡𝛽^1/𝛽𝜕^𝛼𝑢𝜕𝑦^𝛼,𝑢𝑦,0=𝛿𝑦,𝑦∈ℝ,𝑡≥0, (2.39) instead of (1.12). For 𝛼=1 the density of the limiting process reduces to a Cauchy with scale parameter 𝜆^1/𝛽𝑡/𝛽.

3. Space-Fractional Compound Poisson Process

We define now a space-fractional version of the compound Poisson process (which we will indicate hereafter by SFCPP): indeed, its distribution satisfies (2.5), but with integer time derivative and fractional space derivative. We consider the standard CPP 𝑌^(𝛼)(𝑡)=^𝑁(𝑡)_𝑗=1𝑋_𝑗^(𝛼),𝛼∈(0,1], (3.1) where, as usual, 𝑁(𝑡),𝑡>0 is a standard Poisson process with parameter 𝜆 and the random variables 𝑋_𝑗^(𝛼) have the following heavy tail distribution: 𝑓_𝑋(𝛼)(𝑥)=𝜉𝑥^𝛼−1𝐸_𝛼,𝛼−𝜉𝑥^𝛼,𝑥>0,𝛼∈(0,1] (3.2) for 𝜉>0. The Laplace transform of (3.2) is 𝑓_𝑋(𝛼)(𝑘)=𝜉/(𝑘^𝛼+𝜉). The distribution of 𝑋_𝑗^(𝛼) given in (3.2) is usually called Mittag-Leffler and coincides with the geometric-stable law of index 𝛼 (hereafter 𝒢𝒮_𝛼) with parameters 𝜇=0,𝜃=1,and𝜎=[cos(𝜋𝛼/2)/𝜉]^1/𝛼 (see [27]). The density of ∑^𝑛_𝑗=1𝑋_𝑗^(𝛼) is given by 𝑓_𝑋^∗𝑛_(𝛼)𝑦=𝜉^𝑛𝑦^𝛼𝑛−1𝐸^𝑛_{𝛼,𝛼𝑛}−𝜉𝑦^𝛼 (3.3) with Laplace transform 𝑓_𝑋^∗𝑛_(𝛼)(𝑘)=𝜉^𝑛𝑘^𝛼+𝜉^𝑛. (3.4) Note that (3.3) coincides with the density of the 𝑛th event waiting time for the fractional Poisson process 𝒩_𝛼 (see [21]). It is easy to check that the variable 𝑋_𝑗^(𝛼) displays the asymptotic behavior (1.13).

Theorem 3.1. The process 𝑌^(𝛼) defined in (3.1), with 𝑋_𝑗^(𝛼),  𝑗=1,2,…, independent and distributed according to (3.2), has the following distribution: Pr𝑌^(𝛼)(𝑡)≤𝑦=𝑒^−𝜆𝑡1_[0,+∞)𝑦+^𝑦_−∞𝑓_𝑌(𝛼)(𝑧,𝑡)𝑑𝑧,𝑡≥0,𝑦∈ℝ, (3.5) where 𝑓_𝑌(𝛼)=𝑒𝑦,𝑡^−𝜆𝑡𝑦^∞_𝑛=1𝜉𝜆𝑡𝑦^𝛼𝑛𝐸𝑛!^{𝑛𝛼,𝛼𝑛}−𝜉𝑦^𝛼1_(0,+∞)𝑦,𝑡≥0. (3.6) The density (3.6) satisfies the following equation: 𝜉𝜕𝑓𝜕𝑡^𝑌(𝛼)𝜕=−𝜆+𝜕𝜕𝑡𝜕𝑦^𝛼𝛼𝑓_𝑌(𝛼),𝑡≥0,𝑦>0, (3.7) with conditions 𝑓_𝑌(𝛼)𝑦,0=0,_0+∞𝑓_𝑌(𝛼)𝑦,𝑡𝑑𝑦=1−𝑒^−𝜆𝑡. (3.8) The following composition rule holds for the one-dimensional distribution of (3.1): 𝑌^(𝛼)(𝑡)=^𝑑𝒜_𝛼(𝑌(𝑡)), (3.9) where 𝒜_𝛼(𝑡) is the stable subordinator defined in (1.1) and 𝑌 is the standard CPP.

Proof. We start by noting that the absolutely continuous part of the distribution is defined in (0,∞), with the exclusion of 𝑦=0, where only the discrete component gives some contribution.
In order to check (3.7) we evaluate the following fractional derivatives, arguing as in the proof of Theorem 2.1: 𝜕𝑓𝜕𝑡^𝑌(𝛼)𝑦,𝑡=−𝜆𝑒^−𝜆𝑡𝑦^∞_𝑛=1𝜆𝜉𝑡𝑦^𝛼𝑛(𝑛−1)!𝑛!^∞_𝑗=0𝑛+𝑗−1!−𝜉𝑦^𝛼𝑗++𝑒𝑗!Γ𝛼𝑗+𝛼𝑛_−𝜆𝑡𝑦𝑡^∞_𝑛=1𝜆𝜉𝑡𝑦^𝛼𝑛((𝑛−1)!)^2∞_𝑗=0𝑛+𝑗−1!−𝜉𝑦^𝛼𝑗,𝜕𝑗!Γ𝛼𝑗+𝛼𝑛_𝛼𝜕𝑦^𝛼𝑓_𝑌(𝛼)=𝑒𝑦,𝑡^−𝜆𝑡𝑦^∞1+𝛼_𝑛=1𝜆𝜉𝑡𝑦^𝛼𝑛𝑛!(𝑛−1)!^∞_𝑗=0𝑛+𝑗−1!−𝜉𝑦^𝛼𝑗,𝜕𝑗!Γ𝛼𝑗+𝛼𝑛−𝛼_𝛼𝜕𝑦^𝛼𝜕𝑓𝜕𝑡^𝑌(𝛼)𝑒𝑦,𝑡=−^−𝜆𝑡𝑦^∞1+𝛼_𝑛=2𝜆𝜉𝑡𝑦^𝛼𝑛𝑛!(𝑛−2)!^∞_𝑗=0𝑛+𝑗−2!−𝜉𝑦^𝛼𝑗++𝑒𝑗!Γ𝛼𝑗+𝛼𝑛−𝛼_−𝜆𝑡𝑦^1+𝛼𝑡^∞_𝑛=1𝜆𝜉𝑡𝑦^𝛼𝑛((𝑛−1)!)^2∞_𝑗=1𝑛+𝑗−2!−𝜉𝑦^𝛼𝑗++𝑒𝑗−1!Γ𝛼𝑗+𝛼𝑛−𝛼_−𝜆𝑡𝑦^1+𝛼𝑡^∞_𝑛=2𝜆𝜉𝑡𝑦^𝛼𝑛(𝑛−1)!(𝑛−2)!^∞_𝑗=0𝑛+𝑗−2!−𝜉𝑦^𝛼𝑗++𝑒𝑗!Γ𝛼𝑗+𝛼𝑛−𝛼_−𝜆𝑡𝑦^∞1+𝛼_𝑛=1𝜆𝜉𝑡𝑦^𝛼𝑛𝑛!(𝑛−1)!^∞_𝑗=1𝑛+𝑗−2!−𝜉𝑦^𝛼𝑗.𝑗−1!Γ𝛼𝑗+𝛼𝑛−𝛼 (3.10) The initial condition is immediately satisfied by (3.6), while the second condition in (3.8) can be verified as follows: ^∞₀𝑒^−𝑘𝑦𝑓_𝑌(𝛼)𝑦,𝑡𝑑𝑦=𝑒^{∞−𝜆𝑡}_𝑛=1(𝜉𝜆𝑡)^𝑛𝑛!𝑘^∞𝛼𝑛_𝑗=0𝑛+𝑗−1!𝑗!𝑘^𝛼𝑗=𝑒^{∞−𝜆𝑡}_𝑛=11𝑛!𝜉𝜆𝑡𝑘^𝛼+𝜉^𝑛=𝑒^−𝜆𝑡𝑒^{𝜆𝑡𝜉/(𝑘𝛼+𝜉)}−1, (3.11) which, for 𝑘=0, becomes 1−𝑒^−𝜆𝑡. The composition rule given in (3.9) can be verified by taking the Laplace transform of 𝑌^(𝛼), 𝑔_𝑌(𝛼)(𝑘,𝑡)∶=𝔼𝑒^{−𝑘𝑌(𝛼)(𝑡)}=𝑒^{−(𝜆𝑘𝛼/(𝑘𝛼+𝜉))𝑡}, (3.12) which Laplace transformed with respect to 𝑡 gets 𝑔_𝑌(𝛼)(𝑘,𝑠)∶=^∞₀𝑒^−𝑠𝑡𝑔_𝑌(𝛼)()𝑘𝑘,𝑡𝑑𝑡=^𝛼+𝜉𝑘^𝛼()=𝜆+𝑠+𝑠𝜉₀^+∞𝔼𝑒^{−𝑘𝛼𝑌(𝑧)}𝑒^−𝑠𝑧𝑑𝑧. (3.13) Thus, 𝑔_𝑌(𝛼)(𝑘,𝑡)=𝔼𝑒^{−𝑘𝛼𝑌(𝑡)}=₀^+∞𝑒^{−𝑘𝛼𝑣}Pr{𝑌(𝑡)∈𝑑𝑣}, (3.14) so that, by (1.1), we get Pr𝑌^(𝛼)(𝑡)∈𝑑𝑦=₀^+∞ℎ_𝛼𝑦,𝑣Pr{𝑌(𝑡)∈𝑑𝑣}𝑑𝑦, (3.15) and formula (3.9) follows.

Remark 3.2. Equation (3.15) yields an alternative proof of (3.7) noting that the density of 𝒜_𝛼^𝜆,𝜉 satisfies the following equation (where the space-fractional derivative is defined now in the Caputo sense): 𝜕𝑢𝜕𝜕𝑡=−^𝛼𝑢𝜕𝑦^𝛼,𝑢𝑦,0=𝛿𝑦,𝑦,𝑡≥0. (3.16) Indeed, we get 𝜕^𝛼𝜕𝑦^𝛼𝑓_𝑌(𝛼)𝑦,𝑡=₀^+∞𝜕^𝛼𝜕𝑦^𝛼ℎ_𝛼𝑦,𝑣𝑓_𝑌(𝑣,𝑡)𝑑𝑣=−₀^+∞𝜕ℎ𝜕𝑣^𝛼𝑦,𝑣𝑓_𝑌()=𝑣,𝑡𝑑𝑣_0+∞ℎ_𝛼𝜕𝑦,𝑣𝑓𝜕𝑣^𝑌()=(𝜉𝑣,𝑡𝑑𝑣by2.5)=−𝜆𝜕𝜕𝑡₀^+∞ℎ_𝛼𝑦,𝑣𝑓_𝑌()1𝑣,𝑡𝑑𝑣−𝜆𝜕𝜕𝑡₀^+∞ℎ_𝛼𝜕𝑦,𝑣𝑓𝜕𝑣^𝑌()𝜉𝑣,𝑡𝑑𝑣=−𝜆𝜕𝑓𝜕𝑡^𝑌(𝛼)−1𝑦,𝑡𝜆𝜕𝜕𝜕𝑡𝜕𝑦^𝛼𝛼𝑓_Y(𝛼)𝑦,𝑡. (3.17) By considering (3.9) together with (1.2), we can write the following relationship: 𝐹_𝑌^(𝛼)_(𝑡)(𝑧)=Pr𝑌^(𝛼)(𝑡)≤𝑧=Pr𝒜_𝛼(𝑌(𝑡))≤𝑧=Pr𝑌(𝑡)≥ℒ_𝛼(𝑧), (3.18) while for the first version of TFCPP we had, from (2.15), that 𝐹_{𝑌𝛽(𝑡)}(𝑧)=Pr{𝑌(ℒ_𝛽(𝑡))≤𝑧}.
We finally note that the process 𝑌^(𝛼) is still a Markovian and Lévy process, since it is substantially a special case of CPP.

3.1. Special Cases

For 𝛼=1, since the 𝑋_𝑗's reduce to exponential r.v.'s, from (3.6) and (3.7) we retrieve the results (2.11) and (2.13) valid for the standard CPP, under the exponential assumption for 𝑋_𝑗's. As a direct check of (3.9), we can consider the special case 𝛼=1/2, so that the law ℎ_1/2(⋅,𝑧) can be written explicitly as the density of the first passage time of a standard Brownian motion through the level 𝑧>0. Then by considering (3.15) we can write Pr𝑌_1/2(𝑡)∈𝑑𝑦=₀^+∞ℎ_1/2𝑦,𝑣𝑓_𝑌(𝑣,𝑡)𝑑𝑣𝑑𝑦=₀^+∞𝑧𝑒^{−𝑧2/2𝑦}2𝜋𝑦³𝑒^{−𝜉𝑧−𝜆𝑡}𝑧^∞_𝑛=1(𝜆𝜉𝑡𝑧)^𝑛()!=𝑒𝑛!𝑛−1𝑑𝑧𝑑𝑦_−𝜆𝑡𝑦^∞_𝑛=1(𝜆𝜉𝑡)^𝑛()!()𝑛!𝑛−1−1^𝑛𝑑^𝑛𝑑𝜉^𝑛₀^+∞𝑒^{−𝑧2/2𝑦}𝑒2𝜋𝑦^−𝜉𝑧=𝑒𝑑𝑧𝑑𝑦_−𝜆𝑡2𝑦^∞_𝑛=1(𝜆𝜉𝑡)^𝑛()!()𝑛!𝑛−1−1^𝑛𝑑^𝑛𝑑𝜉^𝑛𝐸_1/2,1−𝜉𝑦^1/2𝑑𝑦, (3.19) where the last equality holds by (2.11)-( 2.12) in [28]; then, by (1.10.3) in [12], we get Pr𝑌_1/2(𝑡)=𝑒∈𝑑𝑦^−𝜆𝑡2𝑦^∞_𝑛=1(𝜆𝜉𝑡)^𝑛()!𝑦𝑛−1𝑛!^𝑛/2∞_𝑗=0𝑛+𝑗!−𝜉𝑦^1/2𝑗=𝑒𝑗!Γ𝑗/2+𝑛/2+1𝑑𝑦_−𝜆𝑡𝑦^∞_𝑛=1𝜆𝜉𝑡𝑦^1/2𝑛𝐸𝑛!^{𝑛1/2,𝑛/2}−𝜉𝑦^1/2𝑑𝑦. (3.20)

3.2. Asymptotic Results

We study now the asymptotic behavior of the rescaled version of 𝑌^(𝛼) defined as ℎ𝑌^(𝛼)𝑡𝑟=^{𝑁(𝑡/𝑟)}_𝑗=1ℎ𝑋_𝑗^(𝛼), (3.21) for ℎ,𝑟→0. The Fourier transform of the r.v.'s, 𝑋_𝑗^(𝛼), for any 𝛼∈(0,1), is given by 𝑓_𝑋(𝛼)(𝜅)=11+(1/𝜉)cos(𝜋𝛼/2)|𝜅|^𝛼1−𝑖sgn(𝜅)tan(𝜋𝛼/2) (3.22) (see [27], formula (2.4.1)), which, in the limit, behaves as 𝑓_𝑋(𝛼)(ℎ𝜅)≃1−𝐴ℎ^𝛼|𝜅|^𝛼,ℎ→0, (3.23) where 𝐴=(1/𝜉)cos(𝜋𝛼/2)(1−𝑖sgn(𝜅)tan(𝜋𝛼/2)). Thus, the characteristic function of (3.21) can be written as 𝑔_{ℎ𝑌(𝛼)}𝑡𝜅,𝑟=𝑒^{𝜆(𝑡/𝑟)[}𝑓_{𝑋(𝛼)(ℎ𝜅)−1]}≃𝑒^{−(𝜆𝑡/𝜉)cos(𝜋𝛼/2)|𝜅|𝛼(1−𝑖sgn(𝜅)tan(𝜋𝛼/2))},𝛼∈(0,1), (3.24) for ℎ,𝑟→0, s.t. ℎ^𝛼/𝑟→1.We can conclude that ℎ𝑌^(𝛼)𝑡𝑟⇒𝒜_𝛼^𝜆,𝜉(𝑡), (3.25) where the limiting process is represented, in this case, by an 𝛼-stable subordinator 𝒜_𝛼^𝜆,𝜉(𝑡) with parameters 𝜇=0,  𝜃=1, 𝜎=((𝜆𝑡/𝜉)cos𝜋𝛼/2)^1/𝛼, whose density satisfies 𝜕𝑢𝜆𝜕𝑡=−𝜉𝜕^𝛼𝑢𝜕𝑦^𝛼,𝑢𝑦,0=𝛿𝑦,𝑦>0,𝑡>0. (3.26)

4. Compound Poisson Processes Fractional in Time and Space

We consider now together the results obtained in the previous sections, by defining a CPP fractional both in space and time (STFCPP), that is, 𝑌_𝛽^(𝛼)(𝑡)=^{𝒩𝛽(𝑡)}_𝑗=1𝑋_𝑗^(𝛼),𝑡>0, (4.1) where 𝑋_𝑗^(𝛼)'s are i.i.d. with density (3.2) and 𝒩_𝛽(𝑡), 𝑡>0 is again the FPP.

Theorem 4.1. The process 𝑌_𝛽^(𝛼)(𝑡), 𝑡>0, defined in (4.1) has the following distribution: Pr𝑌_𝛽^(𝛼)(𝑡)≤𝑦=𝐸_𝛽,1−𝜆𝑡^𝛽1_[0,+∞)𝑦+^𝑦_−∞𝑓_𝑌𝛽^(𝛼)(𝑧,𝑡)𝑑𝑧,𝑡≥0,𝑦∈ℝ, (4.2) where 𝑓_𝑌𝛽^(𝛼)=1𝑦,𝑡𝑦^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝛼𝑛𝐸^𝑛+1_{𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽𝐸^𝑛_{𝛼,𝛼𝑛}−𝜉𝑦^𝛼1_(0,+∞)𝑦,𝑡≥0. (4.3) The density 𝑓_𝑌𝛽^(𝛼) solves the following equation: 𝜉𝜕^𝛽𝜕𝑡^𝛽𝑓_𝑌𝛼𝛽𝜕=−𝜆+^𝛽𝜕𝑡^𝛽𝜕^𝛼𝜕𝑦^𝛼𝑓_𝑌𝛽^(𝛼),𝑡≥0,𝑦>0, (4.4) with conditions 𝑓_𝑌𝛽^(𝛼)𝑦,0=0,_0+∞𝑓_𝑌𝛽^(𝛼)𝑦,𝑡𝑑𝑦=1−𝐸_𝛽,1−𝜆𝑡^𝛽. (4.5) The following equality of the one-dimensional distributions holds: 𝑌_𝛽^(𝛼)(𝑡)=^𝑑𝒮_𝛼𝑌_𝛽(𝑡). (4.6)

Proof. In order to check (4.4) we evaluate the following fractional derivatives: 𝜕𝜕𝑡^𝛽𝑓_𝑌𝛽^(𝛼)=1𝑦,𝑡𝑦𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝛼𝑛(𝑛−1)!𝑛!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1^∞_𝑟=0(𝑛+𝑟−1)!−𝜉𝑦^𝛼𝑟(),𝜕𝑟!Γ𝛼𝑟+𝛼𝑛_𝛼𝜕𝑦^𝛼𝑓_𝑌𝛽^(𝛼)=1𝑦,𝑡𝑦^∞1+𝛼_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝛼𝑛𝑛!(𝑛−1)!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗𝑗!Γ𝛽𝑗+𝛽𝑛+1^∞_𝑟=0(𝑛+𝑟−1)!−𝜉𝑦^𝛼𝑟(),𝜕𝑟!Γ𝛼𝑟+𝛼𝑛−𝛼_𝛼𝜕𝑦^𝛼𝜕𝜕𝑡^𝛽𝑓_𝑌𝛽^(𝛼)=1𝑦,𝑡𝑦^1+𝛼𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝛼𝑛𝑛!(𝑛−1)!^∞_𝑗=0𝑛+𝑗!−𝜆𝑡^𝛽𝑗×𝑗!Γ𝛽𝑗+𝛽𝑛−𝛽+1_∞𝑟=0(𝑛+𝑟−1)!−𝜉𝑦^𝛼𝑟().𝑟!Γ𝛼𝑟+𝛼𝑛−𝛼 (4.7) By some algebraic manipulations we finally get (4.4). While the initial condition is trivially satisfied, the second condition in (4.5) can be checked as follows: ^∞₀𝑒^−𝑘𝑦𝑓_𝑌𝛽^(𝛼)𝑦,𝑡𝑑𝑦=^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑛𝑘^𝛼𝑛𝐸^𝑛+1_{𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽∞_𝑟=0𝑟−𝑛+𝑟−1𝜉𝑘^𝛼𝑟=^∞_𝑛=1𝜆𝜉𝑡^𝛽𝑘^𝛼+𝜉^𝑛𝐸^𝑛+1_{𝛽,𝛽𝑛+1}−𝜆𝑡^𝛽=by(2.30)of[21]=𝐸_𝛽,1−𝜆𝜉𝑡^𝛽𝑘^𝛼𝑘^𝛼+𝜉−𝐸_𝛽,1−𝜆𝑡^𝛽, (4.8) which, for 𝑘=0, becomes 1−𝐸_𝛽,1(−𝜆𝑡^𝛽).
The relationship (4.6) can be checked by evaluating the double Laplace transform of 𝑌_𝛽^(𝛼) as follows: 𝑔_𝑌𝛽^(𝛼)(𝑘,𝑠)=₀^+∞𝔼𝑒^{−𝑘𝑌𝛽(𝛼)(𝑡)}𝑒^−𝑠𝑡𝑠𝑑𝑡=^𝛽−1𝑘^𝛼+𝜉𝑠^𝛽𝑘^𝛼+𝜉+𝜆𝑘^𝛼. (4.9) We then rewrite formula (4.9) as 𝑔_𝑌𝛽^(𝛼)(𝑘,𝑠)=𝑠^𝛽−1₀^+∞𝑒^{−𝑠𝛽𝑧}𝔼𝑒^{−𝑘𝛼𝑌(𝑧)}𝑑𝑧 (4.10) and we follow the same lines which lead to (2.15) to get the conclusion.

Remark 4.2. For 𝛼=1 formulae (4.3) and (4.4) coincide with (2.4) and (2.5), while for 𝛼=𝛽=1 we get (3.6) and (3.7).
From (4.6), by considering (1.2), we get the following relation: 𝐹_𝑌𝛽^(𝛼)_(𝑡)(𝑧)=Pr𝑌_𝛽^(𝛼)(𝑡)≤𝑧=Pr𝒮_𝛼𝑌_𝛽(𝑡)≤𝑧=Pr𝑌_𝛽(𝑡)≥ℒ_𝛼(𝑧), (4.11) where ℒ_𝛼 is the inverse stable subordinator.

4.1. Asymptotic Results

For the rescaled version of 𝑌_𝛽^(𝛼) we obtain the following asymptotic result, which agrees with (1.14) and (1.15) proved in [17]: the characteristic function of the process ℎ𝑌_𝛽^(𝛼)𝑡𝑟=^{𝒩𝛽(𝑡/𝑟)}_𝑗=1ℎ𝑋_𝑗^(𝛼) (4.12) can be written as 𝑔_ℎ𝑌𝛽^(𝛼)𝑡𝜅,𝑟=𝐸_𝛽,1𝜆𝑡^𝛽𝑟^𝛽𝑓_{ℎ𝑋(𝛼)}(𝜅)−1. (4.13) By applying formula (3.23) we conclude that (4.13) converges, for ℎ,𝑟→0 s.t. ℎ^𝛼/𝑟^𝛽→1, to 𝐸_𝛽,1−𝜆𝑡^𝛽𝜉|𝜅|^𝛼cos𝜋𝛼21−𝑖sgn(𝜅)tan𝜋𝛼2 (4.14) so that the process ℎ𝑌_𝛽^(𝛼)(𝑡/𝑟) converges weakly to the 𝛼-stable subordinator 𝒜_𝛼^𝜆,𝜉(𝑡), composed with the inverse 𝛽-stable subordinator ℒ_𝛽. Indeed, the characteristic function of 𝒜_𝛼^𝜆,𝜉(ℒ_𝛽(𝑡)) can be evaluated as follows: ₀^+∞𝑒^−𝑠𝑡𝑓_{𝒜𝛼𝜆,𝜉(ℒ𝛽)}(𝜅,𝑡)𝑑𝑡=₀^+∞𝑒^−𝑠𝑡𝑑𝑡₀^+∞𝑒^𝑖𝜅𝑦𝑑𝑦₀^+∞𝑝_𝛼^𝜆,𝜉𝑦;𝑧𝑙_𝛽(𝑧,𝑡)𝑑𝑧=𝑠^𝛽−1₀^+∞𝑒^{−𝑧𝜆|𝜅|𝛼𝐴}𝑒^{−𝑧𝑠𝛽}=𝑠𝑑𝑧_𝛽−1𝜆|𝜅|^𝛼𝐴+𝑠^𝛽, (4.15) where ℎ_𝛼^𝜆,𝜉(𝑦,𝑧) is the law of 𝒜_𝛼^𝜆,𝜉(𝑧) and 𝐴=(1/𝜉)cos(𝜋𝛼/2)(1−𝑖sgn(𝜅)tan(𝜋𝛼/2)). By inverting the Laplace transform in (4.15) we get (4.14). The density of 𝒜_𝛼^𝜆,𝜉(ℒ_𝛽(𝑡)) satisfies the following equation: 𝜕^𝛽𝑢𝜕𝑡^𝛽𝜆=−𝜉𝜕^𝛼𝑢𝜕𝑦^𝛼,𝑦,𝑡>0, (4.16) as can be easily seen from (4.15) (see also [29]). A relevant special case of this result can be obtained by taking 𝛼=𝛽=𝜈, so that the composition 𝒜_𝜈^𝜆,𝜉(ℒ_𝜈(𝑡)) is proved to display a Lamperti-type law (see on this topic [30, 31]); therefore, the latter can be seen as the weak limit of the STFCPP.

We note that in the particular case 𝛽=1, the Fourier transform (4.14) reduces to (3.24) and correspondingly (4.16) coincides with (3.26).

Finally, we consider the case where we have 𝑁_𝛽(𝑡) in place of 𝑁_𝛽(𝑡). If the jumps are Mittag-Leffler distributed, we get the following space-time fractional CPP: 𝑌_𝛼,𝛽(𝑡)=^{𝑁𝛽(𝑡)}_𝑗=1𝑋_𝑗^(𝛼), (4.17) whose distribution is given by Pr𝑌_𝛼,𝛽=1≤𝑦𝐸_𝛽,1−𝜆𝑡^𝛽1_[0,+∞)𝑦+^𝑦_−∞𝑓_{𝑌𝛼,𝛽}(𝑧,𝑡)𝑑𝑧,𝑡≥0,𝑦∈ℝ, (4.18) where 𝑓_{𝑌𝛼,𝛽}=1𝑦,𝑡𝐸_𝛽,1𝜆𝑡^𝛽∞_𝑛=1𝜆𝜉𝑡^𝛽𝑦^𝛼𝑛Γ𝐸𝛽𝑛+1^{𝑛𝛼,𝛼𝑛}−𝜉𝑦^𝛼1_[0,+∞)𝑦,𝑡≥0. (4.19) The rescaled version of (4.17) is defined as ℎ𝑌_𝛼,𝛽𝑡𝑟=^{𝑁𝛽(𝑡/𝑟)}_𝑗=1ℎ𝑋_𝑗^(𝛼)⇒𝒜_𝛼^𝜆,𝜉(𝑡) (4.20) for ℎ,𝑟→0, s.t. ℎ^𝛼/𝑟→1, where again 𝒜_𝛼^𝜆,𝜉 denotes the 𝛼-stable subordinator with characteristic function given in (3.24) (last line). Thus, in the limit, the fractional nature of the counting process 𝑁_𝛽 does not exert any influence, in analogy with the result given in (3.25).

5. Fractional-Difference Compound Poisson Process

We present now a final version of the fractional CPP, where the fractionality of the counting process is referred to the difference operator involved in the recursive equation governing its distribution. Let 𝐵 denote the standard backward shift operator, Δ=1−𝐵, and let 𝛾 be a fractional parameter in (0,1], then the fractional recursive differential equation 𝑑𝑝𝑑𝑡^Δ𝑘(𝑡)=−𝜆^𝛾Δ^𝛾𝑝^Δ_𝑘(𝑡),𝑝^Δ_𝑘(0)=1_[𝑘=0], (5.1) has been introduced in [9]. In (5.1) the following definition of the fractional difference operator Δ^𝛾 of a function 𝑓(𝑛) has been used (see [12], formula (2.8.2), page121): Δ^𝛾𝑓(𝑛)=^∞_𝑗=0(−1)^𝑗𝛾_𝑗𝑓,𝑗!𝑛−𝑗 (5.2) where (𝑥)_𝑛=𝑥(𝑥−1)⋯(𝑥−(𝑛−1)) is the falling factorial. We use the notation 𝑝^Δ_𝑘(𝑡)∶=Pr{𝑁_Δ(𝑡)=𝑘},𝑘≥0,𝑡>0, and we have 𝑝^Δ_𝑘(𝑡)=(−1)^𝑘𝑘!^∞_𝑟=0−𝜆^𝛾𝑡^𝑟𝑟!𝛾𝑟_𝑘,𝛾∈(0,1]. (5.3) It can be proved that 𝑁_Δ is not a renewal process, by verifying that the density of the 𝑘th event waiting time cannot be expressed as 𝑘th convolution of i.i.d. random variables. Nevertheless, 𝑁_Δ(𝑡) is a Lévy process, with infinite expected value for any 𝑡. Moreover, by (5.3), one can check that (as ℎ→0) Pr𝑁_Δ(ℎ)=𝑘=(−1)^𝑘+1𝜆^𝛾𝛾_𝑘(ℎ)𝑘!ℎ+𝑜,∀𝑘≥1 (5.4) instead of 𝑜(ℎ) for 𝑘≥2, as for the standard or the time-fractional Poisson process. We can obtain (5.1) from (5.4) by taking into account that the increments are independent and stationary.

Let us define the corresponding fractional-difference compound Poisson process (hereafter ΔFCPP) as 𝑌_Δ(𝑡)=^𝑁Δ(𝑡)_𝑗=1𝑋_𝑗,𝑡≥0,𝛾∈(0,1] (5.5) so that we can obtain, under the assumption of i.i.d. exponential 𝑋_𝑗's, the distribution of 𝑌_Δ together with the differential equation which is satisfied by its absolutely continuous component.

Theorem 5.1. For 𝛾∈(0,1], the distribution of the process 𝑌_Δ defined in (5.5), with 𝑋_𝑗, 𝑗=1,2,…, independent and exponentially distributed with parameter 𝜉, is given by Pr𝑌_Δ(𝑡)<𝑦=𝑒^{−𝜆𝛾𝑡}1_[0,+∞)𝑦+^𝑦₀𝑓_𝑌Δ(𝑧,𝑡)𝑑𝑧,𝑡,𝑦≥0, (5.6) where 𝑓_𝑌Δ=𝑒𝑦,𝑡^−𝜉𝑦𝑦^∞_𝑛=1−𝜉𝑦^𝑛𝑛!(𝑛−1)!^∞_𝑟=0−𝜆^𝛾𝑡^𝑟𝑟!𝛾𝑟_𝑛1_[0,+∞)𝑦,𝑡≥0. (5.7) The density 𝑓_𝑌Δ solves the differential equation: 𝜉𝐷^1/𝛾_−,𝑡𝑓_𝑌Δ=𝜆−𝐷^1/𝛾_−,𝑡𝜕𝑓𝜕𝑦^𝑌Δ, (5.8) where 𝐷^1/𝛾_0−;𝑡 is the right-sided fractional Riemann-Liouville derivative on the half-axis ℝ⁺, with conditions 𝑓_𝑌Δ𝐷𝑦,0=0,_𝑟−,𝑡𝑓_𝑌Δ𝑦,𝑡_𝑡=0=Φ_𝑟𝑦,_0+∞𝑓_𝑌Δ𝑦,𝑡𝑑𝑦=1−𝑒^{−𝜆𝛾𝑡}, (5.9) where Φ_𝑟(𝑦)=(𝜆^𝛾𝑟𝑒^−𝜉𝑦/𝑦)∑^∞_𝑛=1((−𝜉𝑦)^𝑛(𝛾𝑟)_𝑛/𝑛!(𝑛−1)!).
The following subordinating relationship holds for (5.5): 𝑌_Δ(𝑡)=^𝑑𝑌𝒜_𝛾(𝑡), (5.10) where, as usual, 𝒜_𝛾 denotes the 𝛾-stable subordinator.

Proof. Formula (5.7) can be easily derived by (5.3) and can be checked by verifying that, for 𝛾=1, it reduces to (2.11): 𝑓_𝑌Δ(𝑦,𝑡)_𝛾=1=𝑒^−𝜉𝑦𝑦^∞_𝑛=1−𝜉𝑦^𝑛𝑛!(𝑛−1)!^∞_𝑟=𝑛(−𝜆𝑡)^𝑟(𝑟)𝑟!^𝑛=𝑒^−𝜉𝑦𝑦^∞_𝑛=1−𝜉𝑦^𝑛()!()𝑛!𝑛−1−𝜆𝑡^𝑛𝑒^−𝜆𝑡. (5.11) We now prove the relationship (5.10) as follows. The Laplace transform of 𝑌_Δ(𝑡) is given by 𝑔_𝑌Δ(𝑘,𝑡)=𝔼𝑒^{−𝑘𝑌Δ(𝑡)}=^∞_𝑛=0(−𝜉)^𝑛1𝑛!(𝑘+𝜉)^𝑛∞_𝑟=0−𝜆^𝛾𝑡^𝑟𝑟!𝛾𝑟_𝑛=^∞_𝑟=0−𝜆^𝛾𝑡^𝑟𝑟!^∞_𝑛=0𝛾𝑟_𝑛−𝜉𝑛!𝑘+𝜉^𝑛=^∞_𝑟=0−𝜆^𝛾𝑡^𝑟𝜉𝑟!1−𝑘+𝜉^𝛾𝑟=𝑒^{−𝜆𝛾𝑘𝛾𝑡/(𝑘+𝜉)𝛾}; (5.12) moreover, (5.12) can be rewritten as 𝑔_𝑌Δ(𝑘,𝑡)=₀^+∞𝑒^−𝜆𝑧𝑒^{𝜆𝜉𝑧/(𝑘+𝜉)}ℎ_𝛾()=𝑧,𝑡𝑑𝑧_∞𝑛=0(𝜆𝜉)^𝑛𝑛!(𝜉+𝑘)^𝑛₀^+∞𝑧^𝑛𝑒^−𝜆𝑧ℎ_𝛾()=𝑧,𝑡𝑑𝑧_0+∞𝔼𝑒^{−𝑘𝑌(𝑧)}ℎ_𝛾(𝑧,𝑡)𝑑𝑧, (5.13) which gives (5.10). Thus, we also have that f_𝑌Δ𝑦,𝑡=₀^+∞𝑓_𝑌𝑦,𝑧ℎ_𝛾(𝑧,𝑡)𝑑𝑧. (5.14) In order to prove that (5.7) satisfies (5.8), we recall the following result proved in [32]: the density ℎ_𝜈/𝑛(𝑦,𝑡) of the stable subordinator 𝒜_𝜈/𝑛 is governed by the following equation (as well as by (3.16) for 𝛼=𝜈/𝑛): 𝐷^𝑛_−,𝑡ℎ_𝜈/𝑛=𝜕^𝜈𝜕𝑦^𝜈ℎ_𝜈/𝑛,𝑦,𝑡≥0,𝜈∈(0,1], (5.15) for 𝑛∈ℕ, with conditions ℎ_𝜈/𝑛()ℎ0,𝑡=0,_𝜈/𝑛𝑦,𝐷𝑦,0=𝛿_𝑟−,𝑡ℎ_𝜈/𝑛(𝑦,𝑡)_𝑡=0=𝑦^{−(𝜈𝑟/𝑛)−1}Γ(−𝑟𝜈/𝑛),𝑟=1,…,𝑛−1. (5.16) We can prove that the slightly different result holds: 𝐷^1/𝛾_−,𝑡ℎ_𝛾=𝜕ℎ𝜕𝑦^𝛾(]1,𝑦,𝑡≥0,𝛾∈0,1,𝑛=𝛾+1 (5.17) with the following conditions ℎ_𝛾()ℎ0,𝑡=0,_𝛾𝑦,𝐷𝑦,0=𝛿_𝑟−,𝑡ℎ_𝛾(𝑦,𝑡)_𝑡=0=𝑦^{−𝛾𝑟−1}Γ−𝛾𝑟,𝑟=1,…,𝑛−1. (5.18) Equation (5.17) can be checked by resorting to the Laplace transform with respect to 𝑦 as follows: 𝐷^1/𝛾_−,𝑡ℎ_𝛾(𝑘,𝑡)=𝐷^1/𝛾_−,𝑡𝑒^{−𝑘𝛾𝑡}=by(2.2.15)of[12]=𝑘𝑒^{−𝑘𝛾𝑡}=₀^+∞𝑒^−𝑘𝑦𝜕ℎ𝜕𝑦^𝛾𝑦,𝑡𝑑𝑦. (5.19) Analogously, we can check (5.18): in particular we get 𝐷^𝑟_−,𝑡ℎ_𝛾(𝑘,𝑡)_𝑡=0=(−1)^𝑟𝜕^𝑟𝜕𝑡^𝑟ℎ_𝛾(𝑘,𝑡)_𝑡=0=𝑘^𝛾𝑟=₀^+∞𝑒^−𝑘𝑦𝑦^{−𝛾𝑟−1}Γ−𝛾𝑟𝑑𝑦. (5.20) We now take the derivative of (5.14) of order 1/𝛾 with respect to 𝑡: 𝐷^1/𝛾_−,𝑡𝑓_𝑌Δ𝑦,𝑡=₀^+∞𝑓_𝑌𝑦,𝑧𝐷^1/𝛾_−,𝑡ℎ_𝛾()=𝑧,𝑡𝑑𝑧_0+∞𝑓_𝑌𝜕𝑦,𝑧ℎ𝜕𝑧^𝛾()=(𝑧,𝑡𝑑𝑧byconsidering2.11)=−₀^+∞𝜕𝑓𝜕𝑧^𝑌𝑦,𝑧ℎ_𝛾()=1𝑧,𝑡𝑑𝑧𝜉𝜕𝜕𝑦₀^+∞𝜕𝑓𝜕𝑧^𝑌𝑦,𝑧ℎ_𝛾()𝜆𝑧,𝑡𝑑𝑧+𝜉𝜕𝑓𝜕𝑦^𝑌Δ1𝑦,𝑡=−𝜉𝜕𝐷𝜕𝑦^{1/𝛾−,𝑡}𝑓_𝑌Δ+𝜆𝑦,𝑡𝜉𝜕𝑓𝜕𝑦^𝑌Δ𝑦,𝑡. (5.21) We remark that, for 𝛾=1,  𝐷^1/𝛾_−,𝑡𝑓_𝑌Δ=−𝜕𝑓_𝑌Δ/𝜕𝑡, and therefore the previous equation reduces to (2.13). Finally, we have to check (5.9): the first initial condition is trivially satisfied, while the second condition can be checked either directly by taking the derivatives of (5.7) or by noting that 𝐷^𝑟_−,𝑡𝑓_𝑌Δ𝑦,𝑡=₀^+∞𝑓_𝑌𝑦,𝑧𝐷^𝑟_−,𝑡ℎ_𝛾()=(=𝑒𝑧,𝑡𝑑𝑧by5.18)_−𝜉𝑦𝑦^∞_𝑛=1𝜆𝜉𝑦^𝑛𝑛!(𝑛−1)!₀^+∞𝑒^−𝜆𝑧𝑧^{𝑛−𝑟𝛾−1}Γ=𝜆−𝑟𝛾𝑑𝑧_𝛾𝑛𝑒^−𝜉𝑦𝑦^∞_𝑛=1𝜉𝑦^𝑛()!Γ𝑛!𝑛−1𝑛−𝑟𝛾Γ=𝜆−𝑟𝛾_𝛾𝑛𝑒^−𝜉𝑦𝑦^∞_𝑛=1𝜉𝑦^𝑛𝑛!(𝑛−1)!−𝛾𝑟^(𝑛)=𝜆^𝛾𝑛𝑒^−𝜉𝑦𝑦^∞_𝑛=1−𝜉𝑦^𝑛𝑛!(𝑛−1)!𝛾𝑟_𝑛, (5.22) where, in the last step, we have applied the following relationship between falling and rising factorial (𝑥)_𝑛=(−1)^𝑛(𝑥)^(𝑛). The last condition in (5.9) holds, since 𝑔_𝑌Δ(0,𝑡)=1 by (5.12), so that 𝑓_𝑌Δ(0,𝑡)=𝑔_𝑌Δ(0,𝑡)−𝑒^{−𝜆𝛾𝑡}=1−𝑒^{−𝜆𝛾𝑡}.

Remark 5.2. We show that the distribution of the ΔFCPP satisfies a fractional master equation of order 1/𝛾 greater than one, when the jumps have an arbitrary continuous density 𝑓_𝑋. If we consider the (generalized) density function of 𝑌_Δ(𝑡), 𝑔_𝑌Δ𝑦,𝑡∶=𝑒^{−𝜆𝛾𝑡}𝛿𝑦+𝑓_𝑌Δ𝑦,𝑡,𝑦,𝑡≥0, (5.23) then we get 𝐷^1/𝛾_−,𝑡𝑔_𝑌Δ𝑦,𝑡=𝜆𝑔_𝑌Δ𝑦,𝑡−𝜆^+∞_−∞𝑔_𝑌Δ𝑦−𝑥,𝑡𝑓_𝑋(𝑥)𝑑𝑥, (5.24) which is analogue to (1.6) for the TFCPP 𝑌_𝛽. Indeed, by (5.10), we can write (5.23) as 𝑔_𝑌Δ𝑦,𝑡=₀^+∞𝑔_𝑌𝑦,𝑧ℎ_𝛾(𝑧,𝑡)𝑑𝑧, (5.25) where 𝑔_𝑌(𝑦,𝑡)=𝑒^−𝜆𝑡𝛿(𝑦)+𝑓_𝑌(𝑦,𝑡) and 𝑓_𝑌(𝑦,𝑡) are the density of the standard CPP. By taking the fractional time-derivative of (5.25) we get 𝐷^1/𝛾_−,𝑡𝑓_𝑌Δ𝑦,𝑡=₀^+∞𝑔_𝑌𝑦,𝑧𝐷^1/𝛾_−,𝑡ℎ_𝛾()=(=𝑧,𝑡𝑑𝑧by5.17)_0+∞𝑔_𝑌𝜕𝑦,𝑧ℎ𝜕𝑧^𝛾()=(𝑧,𝑡𝑑𝑧by5.18)=−₀^+∞𝜕𝑔𝜕𝑧^𝑌𝑦,𝑧ℎ_𝛾()=𝑧,𝑡𝑑𝑧bytheKolmogorovmasterequation=𝜆₀^+∞𝑔_𝑌𝑦,𝑧ℎ_𝛾(𝑧,𝑡)𝑑𝑧−𝜆^+∞_−∞𝑓_𝑋(𝑥)₀^+∞𝑔_𝑌𝑦−𝑥,𝑧ℎ_𝛾(𝑧,𝑡)𝑑𝑧𝑑𝑥, (5.26) which coincides with (5.24). For 𝛾=1 (5.24) reduces to the well-known master equation of the standard CPP, by considering again that 𝐷^1/𝛾_−,𝑡𝑓_𝑌Δ=−𝜕𝑓_𝑌Δ/𝜕𝑡.

5.1. Asymptotic Results

We study the asymptotic behavior of the rescaled version of (5.5) under the two alternative assumptions on the r.v.'s representing the jumps: for 𝑋^∗_𝑗 distributed according to (1.9) and for 𝑋_𝑗^(𝛼) with density (3.2). In the first case, we have that ℎ𝑌_Δ𝑡𝑟=^{𝑁Δ(𝑡/𝑟)}_𝑗=1ℎ𝑋^∗_𝑗⇒𝒮_𝛼𝛾(𝑡) (5.27) for ℎ,𝑟→0, s.t. ℎ^𝛼𝛾/𝑟→1, where 𝒮_𝛼𝛾(𝑡) is a symmetric stable process of index 𝛼𝛾 (which is strictly less than one) and parameters 𝜇=0, 𝜃=0, and  𝜎=(𝑡cos𝜋𝛼𝛾/2)^1/𝛼𝛾. Indeed, the characteristic function of (5.27) can be evaluated, by considering that the probability generating function of 𝑁_Δ is 𝐺(𝑢,𝑡)=𝑒^{−𝜆𝛾𝑡(1−𝑢)𝛾} (see [9]) as follows: 𝑓_ℎ𝑌Δ𝑡𝜅,𝑟=𝑒^{−(𝜆𝛾𝑡/𝑟)(1−}𝑓_{∗ℎ𝑋(𝜅))}^𝛾≃𝑒^{−𝜆𝛾𝑡|𝜅|𝛼𝛾}. (5.28) Under the assumption of Mittag-Leffler distributed 𝑋_𝑗^(𝛼)'s, we get instead the following result: the rescaled process ℎ𝑌_Δ^(𝛼)𝑡𝑟=^{𝑁Δ(𝑡/𝑟)}_𝑗=1ℎ𝑋_𝑗^(𝛼)⇒𝒜^𝜆,𝜉_𝛼𝛾(𝑡) (5.29) can be written as 𝑓_ℎ𝑌Δ^(𝛼)𝑡𝜅,𝑟−𝜆≃exp^𝛾𝑡𝜉^𝛾𝑟cos𝜋𝛼2^𝛾ℎ^𝛼𝛾|𝜅|^𝛼𝛾1−𝑖sgn(𝜅)tan𝜋𝛼2^𝛾−𝜆≃exp^𝛾𝑡𝜉^𝛾|𝜅|^𝛼𝛾exp−𝑖sgn(𝜅)𝜋𝛼𝛾2, (5.30) for ℎ,𝑟→0, s.t. ℎ^𝛼𝛾/𝑟→1. The last line of (5.30) corresponds to the Fourier transform of a stable subordinator 𝒜^𝜆,𝜉_𝛼𝛾 of index 𝛼𝛾 and with parameters 𝜇=0,   𝜃=1,  and 𝜎=((𝜆^𝛾𝑡/𝜉^𝛾)cos𝜋𝛼𝛾/2)^1/𝛼𝛾. Therefore, in both cases, the limiting processes are simply the stable symmetric process and the stable subordinator of index 𝛼𝛾, respectively, instead of their compositions with the inverse stable subordinator as happened when 𝒩_𝛽 was used as counting process.

Acronym

CPP:	Compound Poisson process
TFCPP:	Time-fractional compound Poisson process
SFCPP:	Space-fractional compound Poisson process
STFCPP:	Space-time fractional compound Poisson process
ΔFCPP:	Fractional-difference compound Poisson process.

Acknowledgment

The authors thank the referees for the careful reading of the paper and their useful comments.