An asymptotic distribution is given for the partial sums of a stationary time-series with long-range dependence. The law of large numbers for the sample covariance of the series is also derived. The results differ from those given elsewhere in relaxing the assumption of the independence of the innovations of the series.

The existence of ordinary and exponential moments of a point process with conditional intensity of the form is deduced from a Markov chain representation for t – ρN(t). These results form an application of recent theorems of Tweedie (1983a, b) and are used to obtain laws of large numbers for a range of functionals of the process.

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

In this paper some theorems of Dunsmuir and Hannan [2] are corrected. The topological structure of spaces of (likelihood) equivalence classes of arma structures is more closely investigated and its relation to the statistical problem is elucidated. The formulation is also made more general.

From a random sample X1, X2, …, XN there can be constructed N – K + 1 successive sample means of the form for 0 ≦ n ≦ N − K, where Erdös and Rényi (1970) studied the maximum Σ(N, K) of these N – K + 1 sample means. Under appropriate conditions, they showed that for a wide interval of λ's there exist constants C(λ), depending only on λ and the distribution from which the sample was selected, such that Σ(N, [C(λ) log N])→ λ a.s. as N→∞. In the present article, analogous results are developed for the maximum of the N – K + 1 successive sample medians and, more generally, for all sample quantiles.