Let {X(s , t), s = (s 1, s 2) ∈ ℝ2, t ∈ ℝ} be a stationary random field defined over a discrete lattice. In this paper, we consider a set of domain of attraction criteria giving the notion of extremal index for random fields. Together with the extremal-types theorem given by Leadbetter and Rootzen (1997), this will give a characterization of the limiting distribution of the maximum of such random fields.

We extend the results on the extremal properties of chain-dependent sequences considered in Turkman and Walker (1983) by assuming conditions similar to those given by Leadbetter and Nandagopalan (1987) which permit clustering of high values.

The joint limiting distribution of suitably normalized partial sums and maxima in a stationary strong mixing sequence with finite variance is derived. It is found that in the limit the two components are independent. This generalizes Chow and Teugels' result for independent sequences. Motivation for the present study comes from a statistical problem in the analysis of extreme winds.

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let X n(λ) and Y n(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {X n(λ)}, (Y n(λ)}, {I n(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x ], —∞ < x <∞.

It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

A class of quasi-stationary sequences of random variables is introduced. After giving the definition, it is shown that the limiting distributions of the maxima of such sequences, when suitably normalized, converge to one of the three extreme-value distributions.