Score-based generative models driven by $α$-stable L{é}vy processes have been proposed as an alternative to Brownian diffusion models, with empirical evidence suggesting improved sampling and better modeling of heavy-tailed data. Yet theoretical guarantees for these models remain limited. We develop a non-asymptotic convergence analysis for scale-preserving $α$-stable generative dynamics, covering both compactly supported and power-law targets. In p-Wasserstein distance, we quantify the effects of early stopping, initialization, score approximation, and time discretization, and show exponential decay of the initialization error. A comparison with Brownian dynamics reveals that global Wasserstein convergence alone does not fundamentally distinguish the two models: both initialization errors decay exponentially with the forward horizon on the common class of heavy-tailed targets. A sharp separation emerges instead in the tails. At any positive generation time, stable dynamics produces power-law tails with index $α$, whereas Brownian dynamics remains light-tailed. For a target with tail index $ν$, asymptotically correct extreme-quantile scaling is possible precisely when $α$=$ν$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.

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