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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.

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)
Full Text: DOI
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