Collective insurance of weakly dependent risks

The quantity $\Phi$, which is a limit of the ruin probability $P_n$, $n\to\infty$ of a unity of $n$ insurance companies with independent and identically distributed annual claims with unique means and insurance prices $1+b$, $b=n^{\gamma}$, $\gamma>0$, jumps from 0 to 1 when the parameter $\gamma$ passes through some critical value $\gamma^*$. P. Embrechts put a guestion about an investigation of this “phase transition” in a case, when the claims of separate companies are weakly dependent. In this article an exhaustive solution of the question is made. It is based on special model of dependence of separate companies claims. This model represents an insurance claim of separate company as a sum of small random part, common for all companies and proportional to $n^{-\delta}$, $\delta>0$, and finite one, individual for each company. As a result we can get a picture of “the phase transition”, but already in the plane $(\delta,\gamma)$.