We consider a one-dimensional stochastic model of gravitationally interacting adhesive particles with distribution of initial velocities from the domain of normal attraction of a stable law. It is shown that a nonrandom critical time exists if the initial velocities are small enough. Namely, no macroscopic clusters appear before the critical time, while after the critical time, almost all of the mass is concentrated at a single cluster. The order of maximal cluster size for times prior to the critical one is found. If the initial velocities are large enough, then macroscopic clusters appear right after the beginning of system's life, but complete aggregation does not occur within a finite time.
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