Some Compound Fractional Poisson Processes

In this paper, we introduce and study fractional versions of three compound Poisson processes, namely, the Bell-Touchard process, the Poisson-logarithmic process and the generalized P\'olya-Aeppli process. It is shown that these processes are limiting cases of a recently introduced process by Di Crescenzo et al. (2016), namely, the generalized counting process. We obtain the mean, variance, covariance, long-range dependence property etc. for these processes. Also, we obtain several equivalent forms of the one-dimensional distribution of fractional versions of these processes.

Let {Y β (t)} t≥0 be the inverse stable subordinator, that is, the first passage time of a stable subordinator {D β (t)} t≥0 .A stable subordinator {D β (t)} t≥0 is a one-dimensional Lévy process with non-decreasing sample paths.It is known that (see Di Crescenzo et al. where d = denotes equality in distribution and {M(t)} t≥0 is independent of {Y β (t)} t≥0 .Kataria and Khandakar (2022) showed that several recently introduced counting processes such as the Poisson process of order k, Pólya-Aeppli process of order k, Pólya-Aeppli process, negative binomial process etc. are particular cases of the GCP.In Kataria and Khandakar (2021), the authors studied a limiting case of the GFCP, namely, the convoluted fractional Poisson process.
In this paper, we study in detail the fractional versions of BTP, PLP and GPAP.We obtain their Lévy measures.It is shown that these processes are the limiting cases of GCP.We obtain the probability generating function (pgf), mean, variance, covariance etc. for their fractional variants and establish their long-range dependence (LRD) property.Several equivalent forms of the pmf of fractional versions of these processes are obtained.We have shown that the fractional variants of these processes are overdispersed and non-renewal.It is observed that the one-dimensional distributions of their fractional variants are not infinitely divisible and their increments has the short-range dependence (SRD) property.It is shown that these fractional processes are equal in distribution to some particular cases of the compound fractional Poisson process studied by Beghin and Macci (2014).

Preliminaries
In this section, we give some known results and definitions that will be used later.
It reduces to the two-parameter Mittag-Leffler function for γ = 1, and for γ = β = 1 it reduces to the Mittag-Leffler function.

Bell-Touchard process and its fractional version
Here, we introduce a fractional version of a recently introduced counting process, namely, the Bell-Touchard process (BTP) (see Freud and Rodriguez (2022)).
In view of Remark 3.1, we note that several results for BTP can be obtained from the corresponding results for GCP.Next, we present a few of them.
The next result gives a recurrence relation for the pmf of BTP.It follows from Proposition 1 of Kataria and Khandakar (2022).
Proposition 3.1.The state probabilities q(n, t) of BTP satisfy It is known that the GCP is equal in distribution to the following weighted sum of k independent Poisson processes (see Kataria and Khandakar (2022)): where {N j (t)} t≥0 's are independent Poisson processes with intensity λ j 's.On taking λ j = αθ j /j! for all j ≥ 1 and letting k → ∞, we get which agrees with the result obtained by Freud and Rodriguez (2022).As lim t→∞ N j (t)/t = αθ j /j!, j ≥ 1 a.s., we have The next result gives a martingale characterization for the BTP whose proof follows from Proposition 2 of Kataria and Khandakar (2022).
Proposition 3.2.The process M(t) − αθe θ t t≥0 is a martingale with respect to a natural filtration From (1.8), it follows that the BTP is a Lévy process as it is equal in distribution to a compound Poisson process.So, its mean, variance and covariance are given by Cov(M(s), M(t)) = αθ(θ + 1)e θ min{s, t}, respectively.The BTP exhibits the overdispersion property as Var(M(t)) − E(M(t)) = αθ 2 e θ t > 0 for t > 0.
The characteristic function of BTP can be obtained by taking λ j = αθ j /j! for all j ≥ 1 and letting k → ∞ in Eq. ( 12) of Kataria and Khandakar (2022).It is given by So, its Lévy measure is given by µ(dx) = α ∞ j=1 θ j j! δ j dx, where δ j 's are Dirac measures.
Remark 3.2.For fixed s and large t, the correlation function of BTP has the following asymptotic behaviour: . Thus, it exhibits the LRD property.

3.1.
Fractional Bell-Touchard process.Here, we introduce a fractional version of the BTP, namely, the fractional Bell-Touchard process (FBTP).We define it as the stochastic process {M β (t)} t≥0 , 0 < β ≤ 1 whose state probabilities q β (n, t) = Pr{M β (t) = n} satisfy the following system of fractional differential equations: with initial conditions q β (0, 0) = 1 and q β (n, 0) = 0, n ≥ 1.Note that the system of equations (3.5) is obtained by replacing the integer order derivative in (3.1) by Caputo fractional derivative defined in (1.2).Remark 3.3.On taking λ j = αθ j /j! for all j ≥ 1 and letting k → ∞, the System (1.1) reduces to the System (3.5).Thus, the FBTP is a limiting case of the GFCP.Using (3.5), it can be shown that the pgf On taking the Laplace transform in the above equation and using (1.3), we get where Gβ (u, s) denotes the Laplace transform of G β (u, t).Thus, .
Remark 3.6.The system of differential equations that governs the state probabilities of compound fractional Poisson process was obtained by Beghin and Macci (2014).In view of (3.10), the System (3.5) can alternatively be obtained using Proposition 1 of Beghin and Macci (2014).
Next, we obtain the pmf of FBTP and some of its equivalent versions.The solution u 2β (x, t) of (3.8) is given by (see Beghin and Orsingher (2009)): where W ν,γ (•) is the Wright function defined as follows: be the folded solution to (3.8).
Remark 3.7.An equivalent form of the pmf of BTP can be obtained from (1.6).It is given by where Ω n k is given in (1.7).If we use (3.14) in the proof of Theorem 3.2 then we can obtain the following alternate form of the pmf of FBTP: The pmf of TFPP with intensity α e θ − 1 is given by (see Beghin and Orsingher (2010), Eq. (2.5)) Pr{N β (t) = n} = α(e θ − 1)t β n E n+1 β,nβ+1 −α(e θ − 1)t β , n ≥ 0. From (3.10), we have We recall that X i 's are independent of N β (t) in (3.10).Thus, for n ≥ 1, we have where x j is the total number of claims of j units and Again as X i 's are iid, we have where we have used (1.8).On substituting (3.19) in (3.16), we get an equivalent expression for the pmf of FBTP in the following form: The pmf (3.17) can be written in the following equivalent form by using Lemma 2.4 of Kataria and Vellaisamy (2017b): where The FBTP exhibits overdispersion as Var (M β (t)) − E (M β (t)) > 0 for all t > 0.
As 0 < β < 1, it follows that the FBTP has the LRD property.
Remark 3.8.For a fixed h > 0, the increment process of FBTP is defined as It can be shown that the increment process {Z h β (t)} t≥0 exhibits the SRD property.The proof follows similar lines to that of Theorem 1 of Maheshwari and Vellaisamy (2016).
The factorial moments of the FBTP can be obtained by taking λ j = αθ j /j! for all j ≥ 1 and letting k → ∞ in Proposition 4 of Kataria and Khandakar (2022).
)), r ≥ 1 be the rth factorial moment of FBTP.Then, The proof of the next result follows on using (3.2), the self-similarity property of {Y β (t)} t≥0 in (3.7) and the arguments used in Proposition 3 of Kataria and Khandakar (2022).
Proposition 3.4.The one-dimensional distributions of FBTP are not infinitely divisible.Remark 3.9.Let the random variable W 1 be the first waiting time of FBTP.Then, the distribution of W 1 is given by Pr{W 1 > t} = Pr{M β (t) = 0} = E β,1 −α(e θ − 1)t β , which coincides with the first waiting time of TFPP with intensity α(e θ − 1) (see Kataria and Vellaisamy (2017a), Remark 3.3).However, the one-dimensional distributions of TFPP and FBTP differ.Thus, the fact that the TFPP is a renewal process (see Meerschaert et al. (2011)) implies that the FBTP is not a renewal process.

Poisson-logarithmic process and its fractional version
Here, we introduce a fractional version of the PLP.First, we give some additional properties of it.
On taking β = 1, λ j = −λ(1 − p) j /j ln p for all j ≥ 1 and letting k → ∞, the governing system of differential equations (1.1) for GCP reduces to the governing system of differential equations (2.6) of PLP.Thus, the PLP is a limiting case of the GCP.Thus, we note that several results for PLP can be obtained from the corresponding results for GCP.
The next result gives a recurrence relation for the pmf of PLP that follows from Proposition 1 of Kataria and Khandakar (2022).
The pgf (2.7) can be rewritten as Ĝ It follows that the PLP is equal in distribution to a weighted sum of independent Poisson processes, that is, where {N j (t)} t≥0 is a Poisson process with intensity λ j = −λ(1 − p) j /j ln p.Thus, where we have used lim t→∞ N j (t)/t = −λ(1 − p) j /j ln p, j ≥ 1 a.s.In view of (1.4) and (4.1), the pmf of PLP is given by q(n, t) where Using (1.6), the pmf of PLP can alternatively be written as q(n, t) = where Ω n k is given in (1.7).The next result gives a martingale characterization for the PLP whose proof follows from Proposition 2 of Kataria and Khandakar (2022).Let r1 = λ(p − 1)/p ln p and r2 = λ(p − 1)/p 2 ln p. From (1.10), it follows that the PLP is a Lévy process.Its Lévy measure can be obtained by taking λ j = −λ(1 − p) j /j ln p for all j ≥ 1 and letting k → ∞ in Eq. ( 13) of Kataria and Khandakar (2022).It is given by μ where δ j 's are Dirac measures.Its mean, variance and covariance are given by E( M(t)) = r1 t, Var( M(t)) = r2 t, Cov( M(s), M(t)) = r2 min{s, t}.(4.5) 4.1.Fractional Poisson-logarithmic process.Here, we introduce a fractional version of the PLP, namely, the fractional Poisson-logarithmic process (FPLP).We define it as the stochastic process { Mβ (t)} t≥0 , 0 < β ≤ 1, whose state probabilities qβ (n, t) = Pr{ Mβ (t) = n} satisfy the following system of differential equations: with initial conditions qβ (0, 0) = 1 and qβ (n, 0) = 0, n ≥ 1.Note that the system of equations (4.6) is obtained by replacing the integer order derivative in (2.6) by Caputo fractional derivative.
Also, on taking λ = − ln p, the System (4.6) reduces to the system of differential equations that governs the state probabilities of fractional negative binomial process (see Beghin (2015), Eq. ( 66)).
Using (4.6), it can be shown that the pgf Ĝβ (u, with Ĝβ (u, 0) = 1.On taking the Laplace transform in the above equation and using (1.3), we get On taking inverse Laplace transform and using (2.1), we get Remark 4.2.On taking β = 1 in (4.7), we get the pgf of PLP given in (2.7).Also, from (4.7) we can verify that the pmf qβ (n, t) sums up to one, that is, The next result gives a time-changed relationship between the PLP and its fractional variant, FPLP.Theorem 4.1.Let {Y β (t)} t≥0 , 0 < β < 1, be an inverse stable subordinator independent of the PLP { M(t)} t≥0 .Then The proof of Theorem 4.1 follows similar lines to that of Theorem 3.1.
Remark 4.4.In view of (1.10) and (4.8), we note that the FPLP is equal in distribution to the following compound fractional Poisson process: where {N β (t)} t≥0 is a TFPP with intensity λ independent of the sequence of iid random variables {X i } i≥1 .Therefore, it is neither Markovian nor a Lévy process.In view of (4.10), the System (4.6) can alternatively be obtained using Proposition 1 of Beghin and Macci (2014).
Theorem 4.2.The pmf qβ (n, t) = Mβ (t) = n} of FPLP is given by where Proof.From (3.12) and (4.9), we have where q(n, x) is given in (4.3).From this point the proof follows similar lines to that of Theorem 3.2.
Remark 4.5.If we use (4.4) in the proof Theorem 4.2 then we can obtain the following alternate form of the pmf of FPLP: where Ω n k is given in (1.7).From (4.10), we have where Θ k n is given in (3.18).
As X i 's are iid, we have where we have used (1.10).Substituting (4.15) in (4.13), we get an equivalent expression for the pmf of the FPLP in the following form: The pmf (4.14) can be written in the following equivalent form by using Lemma 2.4 of Kataria and Vellaisamy (2017b): where Λ k n is given in (3.22).Thus, we have obtained the five alternate forms of the pmf of FPLP given in (4.11), (4.12), (4.14), (4.16) and (4.17).By using the same arguments as used in Remark 3.9, we conclude that the FPLP is not a renewal process.
The proof of Theorem 4.3 follows similar lines to that of Theorem 3.3.

Generalized Pólya-Aeppli Process and its fractional version
Here, we introduce a fractional version of the GPAP.First, we give some additional properties of it.
The next result gives a recurrence relation for the pmf of GPAP whose proof follows from Proposition 1 of Kataria and Khandakar (2022).
Proposition 5.1.The state probabilities q(n, t) = Pr{ M(t) = n}, n ≥ 1 of GPAP satisfy The pgf (2.9) can be expressed as It follows that the GPAP is equal in distribution to a weighted sum of independent Poisson processes, that is, where {N j (t)} t≥0 is a Poisson process with intensity , in probability, ( where we have used lim t→∞ N j (t 2), we get the corresponding limiting result for the Pólya-Aeppli process (see Kataria and Khandakar (2022), Section 4.3).In view of (5.1), the pmf of GPAP can be obtained from (1.4) and it is given by q where Using (1.6), the pmf of GPAP can alternatively be written as q(n, t) = where Ω n k is given in (1.7).From (1.11), it follows that the GPAP is a Lévy process.Its Lévy measure can be obtained by taking λ ) for all j ≥ 1 and letting k → ∞ in Eq. ( 13) of Kataria and Khandakar (2022).It is given by 5.1.Fractional generalized Pólya-Aeppli process.Here, we introduce a fractional version of the GPAP, namely, the fractional generalized Pólya-Aeppli process (FGPAP).We define it as the stochastic process { Mβ (t)} t≥0 , 0 < β ≤ 1, whose state probabilities qβ (n, t) = Pr{ Mβ (t) = n} satisfy the following system of differential equations: with initial conditions qβ (0, 0) = 1 and qβ (n, 0) = 0, n ≥ 1.Note that the system of equations (5.5) is obtained by replacing the integer order derivative in (2.8) by Caputo fractional derivative.
Remark 5.1.On taking ) for all j ≥ 1 and letting k → ∞, the System (1.1) reduces to the System (5.5).Thus, the FGPAP is a limiting case of the GFCP.Also, on taking r = 1, the System (5.5) reduces to the system of differential equations that governs the state probabilities of fractional Pólya-Aeppli process (see Beghin and Macci (2014), Eq. ( 19)).Using (5.5), it can be shown that the pgf Ḡβ (u with Ḡβ (u, 0) = 1.On taking the Laplace transform in the above equation and using (1.3), we get On taking inverse Laplace transform and using (2.1), we get (5.6) Remark 5.2.On taking β = 1 in (5.6), we get the pgf of GPAP given in (2.9).Also, from (5.6) we can verify that the pmf qβ (n, t) sums up to one, that is, The next result gives a time-changed relationship between the GPAP and its fractional version, FGPAP.Theorem 5.1.Let {Y β (t)} t≥0 , 0 < β < 1, be an inverse stable subordinator independent of the GPAP { M(t)} t≥0 .Then (5.7) The proof of Theorem 5.1 follows similar lines to that of Theorem 3.1.
Remark 5.4.In view of (1.11) and (5.7), we note that the FGPAP is equal in distribution to the following compound fractional Poisson process: where {N β (t)} t≥0 is a TFPP with intensity λ independent of {X i } i≥1 .Therefore, it is neither Markovian nor a Lévy process.In view of (5.9), the System (5.5) can alternatively be obtained using Proposition 1 of Beghin and Macci (2014).(5.10) Proof.From (3.12) and (5.8), we have qβ (n, t) = ∞ 0 q(n, x)ū 2β (x, t)dx, where q(n, x) is the pmf of GPAP (see Jacob and Jose (2018), Eq. ( 3)).From this point the proof follows similar lines to that of Theorem 3.2.where Θ k n is given in (3.18).As X i 's are iid, we have where we have used (1.11).Substituting (5.15)  where Λ k n is given in (3.22).Thus, we have obtained the six alternate forms of the pmf of FGPAP given in (5.10)-(5.12),(5.14), (5.16) and (5.17).On substituting r = 1 in these pmfs, we get the equivalent versions of the pmf of fractional Pólya-Aeppli process.
Remark 5.6.The distribution of the first waiting time W1 of FGPAP is given by Pr{ W1 > t} = Pr{ Mβ (t) = 0} = E β,1 −λt β .By using the same arguments as used in Remark 3.9, we conclude that the FGPAP is not a renewal process.

Proposition 4 . 2 .
The process M(t) − λ(p − 1) p ln p t t≥0 is a martingale with respect to a natural filtration F t = σ M(s), s ≤ t .

Proposition 4 . 3 .
The one-dimensional distributions of FPLP are not infinitely divisible.The proof of Proposition 4.3 follows on using (4.2), the self-similarity property of {Y β (t)} t≥0 in (4.8) and the arguments used in Proposition 3 ofKataria and Khandakar (2022).
Let s > 0 be fixed.The process {X(t)} t≥0 is said to exhibit the LRD property if its correlation function has the following asymptotic behaviour: