An aggregated model is proposed, of which the partial-sum process scales to the Karlin stable processes recently investigated in the literature. The limit extremes of the proposed model, when having regularly-varying tails, are characterized by the convergence of the corresponding point processes. The proposed model is an extension of an aggregated model proposed by Enriquez (2004) in order to approximate fractional Brownian motions with Hurst index $H\in(0,1/2)$, and is of a different nature of the other recently investigated Karlin models which are essentially based on infinite urn schemes.