Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

Abstract

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

“…In this paper we establish a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the process X ( t ) in (1.1) in the large-intensity asymptotic regime, in which the arrival rate is large while the response function and the delay and noise variables are fixed (unscaled), and there is no scaling in time. Shot noise processes have been studied in this asymptotic regime in [2, 8, 10, 27, 29, 30]. In this regime, the limit processes in the FCLT are Gaussian processes of a particular structure (assuming the arrival process satisfies an FCLT with a Brownian limit).…”

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

“…In Ramirez‐Perez and Sering (), a cluster shot noise model is studied where the noises depend on the same “cluster mark” within each cluster. In Pang and Zhou (), the noises are assumed to be weakly dependent satisfying the ρ‐mixing condition, and an FCLT is established in the high‐intensity asymptotic regime.…”