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

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