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## Limit Theorems and Phase Transitions for Two Models of Summation of Independent Identically Distributed Random Variables with a Parameter

Theory of Probability and Its Applications. 2015. Vol. 59. No. 2. P. 222-243.

Grabchak M., Molchanov S.

We consider two models of summation of independent identically distributed random variables with a parameter. The first is motivated by financial applications and the second by contact models for species migration. We characterize the limiting distributions and their bifurcations under different relationships between the parameter and the number of summands. We find that in the phase transition we may get limiting distributions that are quite different from those that come up in standard limit theorems. Our results suggest that these limiting distributions may provide better models, at least for certain aggregation levels. Moreover, we show how the parameter determines at which aggregation levels these models apply.\

Keywords: Phase transitionфазовые переходыmixtures of distributionsheavy and light tailsсмеси распределенийтяжелые и легкие хвосты

Publication based on the results of:

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In the late 80s and 90s, theoretical physicists of
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