Functional limit theorems for a new class of non-stationary shot noise processes

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

“…If Ψ ( s , t ) induces a measure, we would have Ψ ( r , s ) + Ψ ( s , t ) = Ψ ( r , t ) for any t ≥ s ≥ r ≥ 0, but we have this equality if and only if t = s = r ≥ 0. The function Ψ does, however, induce a set function satisfying the following superadditivity property, and therefore, the generalized maximal inequalities in Theorems 5.1 and 5.2 of Pang and Zhou () can be applied.…”

“…A useful model for nonstationary noises is to allow the distribution of the noises to depend on the arrival times of the shots. This model has been studied in Pang and Zhou () for infinite‐server queues with service times dependent on the arrival times, and in Pang and Zhou () for general nonstationary shot noise processes. In Pang and Zhou (), when the arrival process has a Brownian limit, as in the case of a renewal process, the limit process is a nonstationary Gaussian process, which is not self‐similar.…”

“…The proof of weak convergence uses a new sufficient convergence criterion (Theorem 6.1 in the Appendix) developed in Pang and Zhou (), which extends the classical convergence criterion, Theorem 13.5 in Billingsley (). It relies on new maximal inequalities that relax the requirement of having a finite measure as the probability or moment upper bound for the increments of the processes (Theorems 10.3 and 10.4 in Billingsley ).…”

“…It relies on new maximal inequalities that relax the requirement of having a finite measure as the probability or moment upper bound for the increments of the processes (Theorems 10.3 and 10.4 in Billingsley ). It allows the upper bound to be a set function with a superadditivity property (see Definition 6.1 and Theorems 5.1 and 5.2 in Pang and Zhou []). To verify that the convergence criterion applies, we compute the second and fourth moments of the increments of the scaled shot noise processes (Lemmas 3.1 and 3.3), verify that they induce a finite set function that satisfies the superadditivity property (Lemma 3.2), and thus prove the probability bound for the increment of the scaled shot noise processes (Lemma 3.4).…”

“…To verify that the convergence criterion applies, we compute the second and fourth moments of the increments of the scaled shot noise processes (Lemmas 3.1 and 3.3), verify that they induce a finite set function that satisfies the superadditivity property (Lemma 3.2), and thus prove the probability bound for the increment of the scaled shot noise processes (Lemma 3.4). To prove the existence of the process in the space , the classical existence criterion Theorem 13.6 in Billingsley () is also extended in Pang and Zhou () (See Theorem 5.3 therein and Theorem 6.2 in the Appendix). We apply this existence criterion to establish the existence of the limit process in the space .…”