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Some thoughts on stationary processes and linear time series analysis

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde
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Abstract

In this paper we trace the development of the asymptotic analysis of autocorrelations for stationary purely non-deterministic time series. We emphasize the interplay between mathematical requirements and modelling philosophy. We then proceed to extend the theory to the case where only a certain weak form of asymptotic independence of the linear prediction errors is needed rather than the earlier martingale difference or independence requirements.

Keywords

STATIONARY PROCESSESAUTOCORRELATIONSASYMPTOTICS

Information

Type
Part 7 - Stationary Processes and Time Series
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988, pp. 309 - 318
Copyright
Copyright © Applied Probability Trust 1988 

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References

Anderson, T. W. (1970) The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
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.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco (2nd edn, 1976).Google Scholar
Esseen, C.-G. and Janson, S. (1985) On moment conditions for normed sums of independent variables and martingale differences. Stoch. Proc. Appl. 19, 173–182.CrossRefGoogle Scholar
Glynn, P. and Iglehart, D. L. (1988) Steady-state simulation output analysis via standardized time series. Math. Operat. Res. to appear.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.CrossRefGoogle Scholar
Hannan, E. J. (1979) The statistical theory of linear systems. Chapter 2 of Developments in Statistics. Vol. 2, ed. Krishnaiah, P. R. Academic Press, New York.Google Scholar
Hannan, E. J. and Heyde, C. C. (1972) On limit theorems for quadratic functions of discrete time series. Ann. Math. Statist. 43, 2058–2066.CrossRefGoogle Scholar
Hannan, E. J., Krishnaiah, P. R. and Rao, M. M., (eds.) (1985) Handbook of Statistics, Vol. 5. Elsevier, New York.Google Scholar
Heyde, C. C. and Seneta, E. (1972) Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Prob. 9, 235–256.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Mann, H. B. and Wald, A. (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11, 173–220.Google Scholar
Schruben, L. (1983) Confidence interval estimation using standardized time series. Operat. Res. 31, 1090–1108.Google Scholar
Wold, H. (1938) A Study in the Analysis of Stationary Time Series. Almqvist and Wicksell, Uppsala.Google Scholar