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On a random vibration model

Part of: Inference from stochastic processes Limit theorems Survival analysis and censored data Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Dawei Huang  and
N. M. Spencer
Dawei Huang*
Affiliation:
Queensland University of Technology
N. M. Spencer*
Affiliation:
Nottingham Trent University
*
∗Postal address: School of Mathematics, Queensland University of Technology, Brisbane, Q4001, Australia.
∗∗Postal address: Department of Maths, Statistics and O.R., Nottingham Trent University, Burton Street, Nottingham NGl 4BU, UK.
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Abstract

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.

Keywords

FM SIGNALSECOND-ORDER STATIONARITYARMA FORM FOR FM SIGNALNON-LINEAR PREDICTIONBARTLETT FORMULA

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G25: Prediction theory 60G35: Signal detection and filtering 62M10: Time series, auto-correlation, regression, etc. 62M20: Prediction 62N99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1141 - 1158
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] An, H. Z., Chen, Z. G. and Hannan, E. J. (1982) Autocorrelation, autoregression and autoregressive approximation. Ann. Statist. 10, 926–936.Google Scholar
[2] Brillinger, D. R. (1981) Time Series: Data Analysis and Theory , Holden-Day, New York.Google Scholar
[3] Chow, Y. S. (1965) Local convergence of martingales and the law of large numbers. Ann. Math. Statist. 36, 552–558.Google Scholar
[4] Doob, J. L. (1994) The elementary Gaussian processes. Ann. Math. Statist. 15, 229–282.Google Scholar
[5] Gordin, M. I. (1969) The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176.Google Scholar
[6] Hall, P and Heyde, C. C. (1980) Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
[7] Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
[8] Huang, D. (1994) Statistical modelling and inference for sinusoids with time varying, random phase and amplitude. Tech. Report. submitted for publication.Google Scholar
[9] Huang, D. (1994) On stochastic FM models. Tech. Report. submitted for publication.Google Scholar
[10] Kelly, C. N. and Gupta, S. C. (1972) Discrete-time demodulation of continuous-time signals. IEEE Trans. Inf. Theory IT-18, 488–493.CrossRefGoogle Scholar
[11] Kolmogoroff, A. (1941) Stationary sequence in Hilbert space. Bull. Math. Univ. Moscow 2, No. 6.Google Scholar
[12] Kumar, B. K. (1992) Novel multireceiver communication systems configurations based on optimal estimation theory. IEEE Trans. Commun. 40, 1767–1780.Google Scholar
[13] Loève, M. (1977) Probability Theory. 4th edn. Springer, Berlin.Google Scholar
[14] Middleton, D. (1987) Introduction to Statistical Communication Theory. Peninsula Publishing, Los Altos, CA.Google Scholar
[15] Papoulis, A. (1968) Systems and Transforms with Applications in Optics. McGraw-Hill, New York.Google Scholar
[16] Rabiner, L. R. and Schafer, R. W. (1978) Digital Processing of Speech Signals. Prentice-Hall, New York.Google Scholar
[17] Shannon, C. E. (1948) A mathematical theory of communication. Bell Sys. Tech. J. 27, 379–423, 623-656.Google Scholar
[18] Taub, H. and Schilling, D. L. (1986) Principles of Communication Systems. 2nd edn. McGraw-Hill, New York.Google Scholar