  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.
