Functional limit theorems for a new class of non-stationary shot noise processes
Guodong Pang and
Stochastic Processes and their Applications, 2018, vol. 128, issue 2, 505-544
We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property.
Keywords: Shot noise processes; Functional weak law of large numbers; Functional central limit theorem; Gaussian limit; Non-stationarity; Skorohod j1 topology; Weak convergence; Maximal inequalities; Criterion of existence in the space D (search for similar items in EconPapers)
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