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The insurance model when the amount of claims depends on the state of theinsured person (healthy, ill, or dead) and claims are connected in a Markovchain is investigated. The signed compound Poisson approximation is applied tothe aggregate claims distribution after $n\in \mathbb {N}$ periods. Theaccuracy of order $O(n^{-1})$ and $O(n^{-1/2})$ is obtained for the local anduniform norms, respectively. In a particular case, the accuracy of estimates intotal variation and non-uniform estimates are shown to be at least of order$O(n^{-1})$. The characteristic function method is used. The results can beapplied to estimate the probable loss of an insurer to optimize an insurancepremium.

Asymptotics for the sum of three state Markov dependent random variables