We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean $\overline{a}$ and variance $v_a$. These two control parameters determine if the avalanche size tends to a stationary distribution (finite scale statistics with finite mean and variance, or power-law tailed statistics with exponent $\in (1,3]$), or instead to a nonstationary regime with log-normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by mathematical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions, and they provide a precise definition of the boundaries between the three regimes.

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• Growth processes
• Irreversible processes