Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

Abstract

We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous P olya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.

Generalized fractional calculus and some models of generalized counting processes

“…therefore, we obtain formula (14). Now we use the fact, that ˜ f (t, λ) is an eigenfunction of the operator D f t (see (10)), from which we conclude that G f (u, t), being given by ˜ f (t, λ(1 − u)), satisfies equation (15). Remark 1.…”

“…) is an eigenfunction of D f t with the eigenvalue ψ(λ(1 − u)) (see (10)), it follows that G ψ,f (u, t) satisfies equation (28). Remark 2.…”

“…Equations ( 30) and (31) were stated in paper [1], where the particular representation of the operator ψ (λ (I − B)) was used, and the equations for the probabilities of time-changed processes were stated using the interplay between the operator ψ (λ (I − B)) and the fractional operator D β t in the Fourier domain. The approach in [1] can be extended for the case where we deal with the generalized fractional operator D f t , as soon as we know that its eigenfunction is given by ˜ f (t, λ) (see (10)). We refer for more detail on this approach to [1].…”

“…, λ k . Note that GCP comprises as particular cases such important for applications models as the Poisson process of order k and Pólya-Aeppli process of order k (see, e.g., [11,10]).…”

“…and their fractional extensions. We refer to papers [7,15,6,10,12,11,13], to mention only few, see also references therein.…”