We show that a theory for stopped two-dimensional random walks is well suited to describe cumulative shock models. Limit theorems for the lifetime/failure time of a system are provided.

The least squares estimators βi(N), j = 1, …, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered when the disturbances have a distribution attracted to a stable law of index α < 2. It is shown that N1/δ(βi(N) – β) converges almost surely to zero for any δ > α. Some comments are made on alternative definitions of the βi(N).

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n−1 Σi=1nXi − μ| > ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) < ∞ if and only if EX12 < ∞. Here it is shown that if EX12 < ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.

Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier.

Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency θ0, is in fact 0 or <it is shown that the estimate, , satisfies for N ≧ N0 (ω) where N0 (ω) is an integer, determined by the realisation, ω, of the process, that is almost surely finite.

A general multivariate non-linear regression model is considered, including as special cases linear regression when the regression matrix is of less than full rank, simultaneous equations systems and regression on an unobservable predetermined variable. Given a time-series of observations at unit intervals we consider the estimation of the parameters, subject to non-linear constraints, by minimizing a criterion based on the Fourier-transformed model. We allow the residuals to be generated by a stationary, linear, process and establish asymptotic properties of our estimates.

This paper deals with the simple Galton-Watson process with immigration, {Xn} with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 < m ≡ F'(1–) < 1), and that 0 < λ ≡ B'(1–) < ∞, 0 < B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1–m)–1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales; and discusses relation of the above theory to that of a first order autoregressive process.