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Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {X ij , j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n 1,…,n k ) = ∑i=1 k ∑j=1 n i X ij and random sums S(k; t) = ∑i=1 k ∑j=1 N i (t)X ij , where N i (t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.

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Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

Negative binomial sums of random variables and discounted reward processes

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2 results

Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

Negative binomial sums of random variables and discounted reward processes