The central limit theorem for time series regression

The central limit problem is considered for a simple regression, where the residuals, x(n), are stationary and the sequence regressed on y^(N)(n), may depend on the number of observations, N, to hand. Two situations are considered, one where the residual is generated by a linear process (i.e. the best linear predictor is the best predictor) and the more general situation where that is not so. Two types of conditions are needed, the first of which limits the contribution of any individual y^(N)(n) and the second of which relates to the mixing properties of x(n). If ε(n) is the linear innovation sequence, in the linear case, with $\text{lim}{}_{\mathrm{k}}\text{→∞}\text{E}\text{(ε(n)}{}^2\text{∥}\text{F}{}_{\mathrm{n?k}}\text{)=}\text{F}{}^2\text{,}\text{F}{}_{\mathrm{n}}$ being the associated family of o-algebra, then the central limit theorem holds under minimal conditions on y^(N)(n). Under sligthly stronger conditions on y^(N)(n) and for x(n) weakly mixing this theorem and associated theorems, are shown to hold under further fairly weak conditions on the dependence of x(n) on its past.