Structural, continuity, and asymptotic properties of a branching particle system

Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions

“…In addition, they are assumed to be independent of the Poisson counting process {Π ρ (t), t ≥ 0} with intensity ρ > 0. In view of [9], [26], [28], the marginals of both these classes of stochastic processes belong to the two-parameter Pólya-Aeppli family of distributions described in Definition 3.1 of Section 3 (see also formulas (5.3)-(5.5) of Section 5). This is an important common feature of these two classes of stochastic processes.…”

“…The BPS's, whose new properties are derived in Section 4, are identical to those dealt with in [9]. Since [9,Introduction] already contains a comprehensive bibliography on relevant BPS's as well as on their applications and limits, we will now provide a few references on the compound Poisson-geometric Lévy processes.…”

“…Since [9,Introduction] already contains a comprehensive bibliography on relevant BPS's as well as on their applications and limits, we will now provide a few references on the compound Poisson-geometric Lévy processes. They were considered as early as in [5, p. 96].…”

“…It is known (see, for example, [26,Section 3], [9,Section 3]) that the totality of the class of Pólya-Aeppli distributions can be parameterized in a manner that this family would constitute an additive exponential dispersion model (or EDM) on the set Z + := {0, 1, 2, ...} (compare [9,Section 3]). We refer to [12,Chapter 3] for a comprehensive consideration of such structures.…”

“…First, it is interesting in its own right, since this class represents a toy example for which numerous approximations can be derived in closed form. Secondly, members of the Pólya-Aeppli class frequently emerge in applications (see, for example, [9,Introduction] and the references therein). Recall that one of them is their use for describing the stochastic evolution of some BPS's.…”