From infinite urn schemes to self-similar stable processes

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We investigate the randomized Karlin model with parameter β∈(0,1), which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index β∕2∈(0,1∕2). We show here that when the randomization is heavy-tailed with index α∈(0,2), then the odd-occupancy process scales to a (β∕α)-self-similar symmetric α-stable process with stationary increments.