Computing the extremal index of special Markov chains and queues (Q1382480)

The paper contributes to the asymptotic of the extremal value distribution. The authors consider the maximum \(M_k=\max \{X_1,X_2,\dots ,X_k\}\) of the stationary Markov chain \(\{X_k\}_{k\in N}\) and take care of the extremal index, i.e. the number \(\theta _1\in [0,1]\) fulfilling \([P(M_n\leq u_n)-F(u_n)^{n\theta _1}]\to 0\) if \(n\to +\infty \) and \(n(1-F(u_n))\to \tau >0\), where \(F\) denotes the stationary distribution of the considered Markov chain. The paper presents assumptions under which the extremal index exists and admits an expression based on a random walk distribution.