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Large claims approximations for risk processes in a Markovian environment. (English) Zbl 0814.60067
The paper provides some results concerning a risk process with initial reserve $$u$$, which evolves in a finite Markovian environment with initial state $$i$$. Denoting by $$\beta_ j$$ the arrival intensity and by $$B_ j$$ the claim size distribution when the environment is in the state $$j( \in E)$$, the authors suppose that there is a subset $$E^{(1)}$$ of $$E$$ such that, for every $$j \in E^{(1)}$$, $$B_ j$$ verify the asymptotic condition $$1-B_ j(x) \sim b_ j (1 - H(x))$$, as $$x \to \infty$$, for some $$b_ j \in (0, + \infty)$$ and some p.d.f. $$H$$, whose tail is a subexponential density, and, for every $$j \in E \backslash E^{(1)}$$, $$1 - B_ j(x) = o(1 - H(x))$$. Under these hypotheses, denoting by $$\psi_ i (u)$$ the probability of ruin for the above mentioned risk process, the authors prove that $$\psi_ i (u) \sim c_ i \int^ \infty_ u (1 - H(x))dx$$, for some explicitly computed constant $$c_ i$$. In the last part of the paper it is proved that similar results hold for the tail of the waiting time in a Markov-modulated $$M/G/1$$ queue whenever the service times satisfy similar hypotheses.

##### MSC:
 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60K25 Queueing theory (aspects of probability theory)
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