On Limit Theorems for Quadratic Functions of Discrete Time Series
In this paper it is shown how martingale theorems can be used to appreciably widen the scope of classical inferential results concerning autocorrelations in time series analysis. The object of study is a process which is basically the second-order stationary purely non-deterministic process and contains, in particular, the mixed autoregressive and moving average process. We obtain a strong law and a central limit theorem for the autocorrelations of this process under very general conditions. These results show in particular that, subject to mild regularity conditions, the classical theory of inference for the process in question goes through if the best linear predictor is the best predictor (both in the least squares sense).
KeywordsTime Series Limit Theorem Classical Theory Central Limit Central Limit Theorem
- 5.Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, Philadelphia.Google Scholar
- 8.Heyde, C. C and Seneta, E. (1972). Estimation theory for growth and immigration rates in a muliplicative process. To appear in J. Appl. Probability.Google Scholar