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 $\alpha ,{\alpha }^{\prime }>0$, 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 $\alpha <{\alpha }^{\prime }\beta$, noise-dominance regime $\alpha >{\alpha }^{\prime }\beta$, and critical regime $\alpha ={\alpha }^{\prime }\beta$. 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.

29 pages. Major revision with a new title

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