Abstract
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).
Received August 7, 1971; revised April 7, 1972.
Chapter PDF
References
Anderson, T. W. and Walker, A. M. (1964). On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. Ann. Math. Statist. 35 1296–1303.
Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis Forecasting and Control. Holden-Day, San Francisco.
Brown, B. M.(1971). Martingale central limit theorems. Ann. Math. Statist. 42 59–66.
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, Philadelphia.
Hannan, E. J. (1970). Multiple Time Series. Wiley, New York.
Hannan, E. J. (1971). Non linear time series regression. J. Appl. Probability 8 767–780.
Heyde, C. C and Seneta, E. (1972). Estimation theory for growth and immigration rates in a muliplicative process. To appear in J. Appl. Probability.
Katznelson, Y. (1968). Harmonic Analysis. Wiley, New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer New York
About this chapter
Cite this chapter
Hannan, E.J., Heyde, C.C. (2010). On Limit Theorems for Quadratic Functions of Discrete Time Series. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_28
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5823-5_28
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-5822-8
Online ISBN: 978-1-4419-5823-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)