Clusters of extremes: modeling and examples
Let \((R_{i},X_{i})\), \(i=1,2,\dots \), be a two-dimensional time series, where a stationary random sequence \( \left\{ R_{n} \right\}_{n \geq 1} \) and \( M_{n}= \max \left\{ R_{1},\ldots,R_{n}\right\}\) could be interpreted as observations and \( \left\{ X_{n} \right\}_{n \geq 1} \) could be interpreted as the time between consecutive observations. The author studies the inter-cluster size \[ T_{1}\left( u \right) =\min\left\{ j \geq 1 : M_{1,j} \leq u, R_{j+1} >u \left| R_{1} > u\right. \right\}, \] the cluster size \[ T_{2}\left( u \right) =\min\left\{ j \geq 1 : L_{1,j} > u, R_{j+1} \leq u | R_{1} \leq u \right\} \] and the return interval between clusters \[ S_{T_{1}\left(u \right) } = \sum_{i=1}^{T_{1}\left(u \right)} X_{i}, \] where \( M_{1,j}=\max\left\{ R_{2},\dots,R_{j}\right\} \), \( M_{1,1}=-\infty \), \( L_{1,j}= \min\left\{ R_{2},\dots,R_{j}\right\} \), \( L_{1,1}=+\infty \). From the author's abstract: ``We derive asymptotic distributions of the cluster and inter-cluster sizes for processes with the extremal index equal to zero, the asymptotic expectation of the inter-cluster size and an exponential rate of convergence of the distribution tail of the return interval between clusters to the stable distribution tail. Distributions of the cluster and inter-cluster sizes of ARMAX, MM and AR(1) processes are obtained.''
scientific article; zbMATH DE number 6829869
scientific article