Phase transition for extremes of a stochastic model with long-range dependence and multiplicative noise

Abstract

We consider a stochastic process with long-range dependence perturbed by multiplicative noise. The marginal distributions of both the original process and the noise have regularly-varying tails, with tail indices $\alpha,\alpha'>0$, respectively. The original process is taken as the regularly-varying Karlin model, a recently investigated model that has long-range dependence characterized by a memory parameter $\beta\in(0,1)$. We establish limit theorems for the extremes of the model, and reveal a phase transition. In terms of the limit there are three different regimes: signal-dominance regime $\alpha<\alpha'\beta$, noise-dominance regime $\alpha>\alpha'\beta$, and critical regime $\alpha = \alpha'\beta$. As for the proof, we actually establish the same phase-transition phenomena for the so-called Poisson--Karlin model with multiplicative noise defined on generic metric spaces, and apply a Poissonization method to establish the limit theorems for the one-dimensional case as a consequence.