Functional limit theorems for shot noise processes with weakly dependent noises

Guodong Pang, Yuhang Zhou

Research output: Contribution to journalArticlepeer-review

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 thedependenceamongthenoises.Themodeland 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 processbyexploitingtheρ-mixing condition. To prove the weak convergence, we es-tablish 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.

Original languageEnglish (US)
Pages (from-to)99-123
Number of pages25
JournalStochastic Systems
Volume10
Issue number2
DOIs
StatePublished - Jun 2020

All Science Journal Classification (ASJC) codes

  • Management Science and Operations Research
  • Statistics, Probability and Uncertainty
  • Modeling and Simulation
  • Statistics and Probability

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