Extremes and clustering of nonstationary max-AR(1) sequences
Let \(\{Y_ i, i \geq 1\}\) be a sequence of i.i.d. random variables and let \(\{Z_ i, \;i \geq 1\}\) be a sequence of independent random variables \((P \{0 \leq Z_ i \leq 1\} = 1\) for all \(i\)'s), independent of \(\{Y_ i\}\). Define the nonstationary max-autoregressive sequence \(\{X_ i, i \geq 1\}\) by \[ X_ i = \begin{cases} X_ 0 \quad & \text{if } i = 0,\\ Z_ i \max \{X_{i - 1}, Y_ i\} & \text{if } i \geq 1, \end{cases} \] with any random variable \(X_ 0\). The authors study the limiting behaviour of the maximum \(M_ n = \max \{X_ i, 0 \leq i \leq n\}\) and evaluate the extremal index for the case where the marginal distribution of \(Y_ i\) is regularly varying at \(\infty\). They also prove that under some additional conditions the point process of exceedances of a boundary \(u_ n\) by \(X_ i\), \(i \leq n\), converges to a compound Poisson process, where the compounding distribution is geometric.
nonstationary max-autoregressive sequence
scientific article; zbMATH DE number 757557
scientific article