On the complete monotonicity of the compound geometric convolution with applications in risk theory

We prove that the complete monotonicity is preserved under mixed geometric compounding, and hence show that the ruin probability, the Laplace transform of the ruin time, and the density of the tail of the joint distribution of ruin and the deficit at ruin in the Sparre Andersen model are completely monotone if the claim size distribution has a completely monotone density.

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