The spectral factorization problem was solved in Hallin (1984) for the class of (non-stationary) m-variate MA(q) stochastic processes, i.e. the class of second-order q-dependent processes. It was shown that such a process generally admits an infinite (mq(mq +1)/2-dimensional) family of possible MA(q) representations. The present paper deals with the invertibility properties and asymptotic behaviour of these MA(q) models, in connection with the problem of producing asymptotically efficient forecasts. Invertible and borderline non-invertible models are characterized (Theorems 3.1 and 3.2). A criterion is provided (Theorem 4.1) by which it can be checked whether a given MA model is a Wold–Cramér decomposition or not; and it is shown (Theorem 4.2) that, under mild conditions, almost every MA model is asymptotically identical with some Wold–Cramér decomposition. The forecasting problem is investigated in detail, and it is established that the relevant invertibility concept, with respect to asymptotic forecasting efficiency, is what we define as Granger-Andersen invertibility rather than the classical invertibility concept (Theorem 5.3). The properties of this new invertibility concept are studied and contrasted with those of its classical counterpart (Theorems 5.2 and 5.4). Numerical examples are also treated (Section 6), illustrating the fact that non-invertible models may provide asymptotically efficient forecasts, whereas invertible models, in some cases, may not. The mathematical tools throughout the paper are linear difference equations (Green's matrices, adjoint operators, dominated solutions, etc.), and a matrix generalization of continued fractions.

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.