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Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities. (English) Zbl 0942.60034
Under the assumption that the jumps are iid and follow a subexponential distribution $$F$$, three stochastic models are discussed: For the Lindley process $$(W_n)$$ given by the recursion $$W_{n+1}=(W_n + X_n)^+$$ let $$M_{\tau}$$ be the maximum within a regenerative cycle with mean $$\mu$$ and let $$G(x)= P(M_{\tau}\leq x)$$. It is shown that the assumption of subexponentiality results in the asymptotics $${\overline G}(x) \sim \mu{\overline F}(x)$$ (which implies that the extremal index $$\theta$$ of $$(W_n)$$ is zero) and $$\|P(\max_{0\leq k\leq n} W_k \leq x) - G^{n/\mu}(x) \|\rightarrow 0$$, $$n\rightarrow \infty$$. If in addition $$F$$ belongs to the max-domain of attraction of an extreme value df $$H$$, then the point process of the exceedances, properly normalized, converges in distribution to a compound Poisson process with intensity $$-\log H$$ and a Pareto compounding distribution. Similar results are obtained for a storage process $$(V_t)$$ which moves between the heavy-tailed jumps downwards according to the ODE $${\dot x}(t)=-r(x(t))$$ where $$r(x)$$ is the release rate at level $$x$$: the maximum of $$(V_t)$$ up to time $$T$$ behaves like the maximal jump in $$[0,T]$$. The tail of the stationary distribution is also found. For a risk process with premium rate $$r(x)$$ and subexponential claims the asymptotic distribution of the ruin time $$\rho (x)$$ is determined: the conditional distribution of $$\rho (x)$$ given $$\rho (x) < \infty$$ converges to the exponential distribution.
Reviewer: E.Pancheva (Sofia)

##### MSC:
 60G70 Extreme value theory; extremal stochastic processes 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.) 60K25 Queueing theory (aspects of probability theory)
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