Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.

Records from are analyzed, where {Yj } is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

We consider the relative error of a tail function when this is approximated by y–α using an estimator of Hill's for α. The results combine recent work of Davis and Resnick on tail estimation with Anderson's work on large deviations in extreme-value theory. Treating separately the domains of attraction of Φα and Λ, we obtain general conditions for the relative error to tend to 0 as u →∞, y → ∞ simultaneously. The results serve as warning against the automatic extrapolation of estimates based on extreme-value approximations.

Let be the kth largest among X n1 , …, Xn [nt], where Xni = (Xi – an )/bn , {Xi } is a sequence of independent random variables and bn > 0 and an are norming constants. Suppose that for each converges in distribution. Then all the finite-dimensional laws of converge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.