= Poisson superposition of insurance portfolios
{c}
{title2=$\lambda=\sum_i\lambda_i,\qquad F=\sum_i\frac{\lambda_i}{\lambda}F_i$}
Independent claim <Poisson processes> of rates $\lambda_i$ merge into a <Poisson process> of rate $\lambda=\sum_i\lambda_i$, by the <Superposition theorem for Poisson point processes>. The merged claim law is a <mixture distribution> of the individual claim laws with weights $\lambda_i/\lambda$. Equivalently, multiplication of the individual compound-Poisson transforms produces $\exp(\lambda[\sum_i(\lambda_i/\lambda)M_i(r)-1])$.
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