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**Extremal theory for stochastic processes.**
*(English)*
Zbl 0648.60039

The authors present a review of the principal results in and related to the distributional theory of extremes of stationary sequences and processes. The review is given in three areas: extremes of sequences of independent, identically distributed random variables, extremes of stationary sequences and extremes of stationary continuous-parameter processes. Significant ideas and methods are described rather than details.

In particular, the centrality of convergence results for point processes associated with extremes (such as exceedances and upcrossings) is emphasized. In many cases the details may be found in the book by the authors and G. Lindgren, Extremes and related properties of random sequences and processes. (1983; Zbl 0518.60021). Applications are given to particular classes of processes (e.g., normal sequences and processes, regenerative and Markov sequences, moving averages, diffusion processes), and connections with the central limit problem of convergence of sums to nonnormal stable distributions are indicated.

In particular, the centrality of convergence results for point processes associated with extremes (such as exceedances and upcrossings) is emphasized. In many cases the details may be found in the book by the authors and G. Lindgren, Extremes and related properties of random sequences and processes. (1983; Zbl 0518.60021). Applications are given to particular classes of processes (e.g., normal sequences and processes, regenerative and Markov sequences, moving averages, diffusion processes), and connections with the central limit problem of convergence of sums to nonnormal stable distributions are indicated.

Reviewer: W.Dziubdiela

### MSC:

60G10 | Stationary stochastic processes |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G15 | Gaussian processes |

60G17 | Sample path properties |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60F05 | Central limit and other weak theorems |