Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities
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.
scientific article; zbMATH DE number 1319919
scientific article