We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb {N} \to \mathbb {N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty $ , that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$ . This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time, real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest ‘mixing’ conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.

We extend large exceedence results for i.i.d. -valued random variables to a class of uniformly recurrent Markov-additive processes and stationary strong-mixing processes. As in the i.i.d. case, the results are proved via large deviations estimates.

In this paper we connect various topological and probabilistic forms of stability for discrete-time Markov chains. These include tightness on the one hand and Harris recurrence and ergodicity on the other. We show that these concepts of stability are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets (‘petite sets').

We use a discrete formulation of Dynkin's formula to establish unified criteria for these stability concepts, through bounding of moments of first entrance times to petite sets. This gives a generalization of Lyapunov–Foster criteria for the various stability conditions to hold. Under these criteria, ergodic theorems are shown to be valid even in the non-irreducible case. These results allow a more general test function approach for determining rates of convergence of the underlying distributions of a Markov chain, and provide strong mixing results and new versions of the central limit theorem and the law of the iterated logarithm.

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.

Records from the sequence Yn = Xn + cn, n ≧ 1 are analyzed, where {Xn} is a strictly stationary random sequence. We prove various limit theorems for the record rate, record times, and record values. The situation when {Xn} is a stationary Gaussian process is considered with special attention given to Gaussian ARMA sequences. Data for the 400 and 800 metre races are used to illustrate these results.

The concept of strong mixing is used to obtain a generalization of results on the asymptotic distribution of finite-parameter estimates of linear processes and extend them for stationary sequences and random fields.

Certain first-order autoregressive processes are shown not to be strong mixing. A direct proof is given. This proof gives considerably more insight into the nature of the result than do proofs by contradiction. The result and proof help to clarify the relation between the autoregressive and strong mixing conditions.