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Stochastic approximation with non-additive measurement noise

Part of: Sequential methods

Published online by Cambridge University Press:  14 July 2016

Han-Fu Chen
Han-Fu Chen*
Affiliation:
The Chinese Academy of Sciences
*
∗Postal address: Laboratory of Systems & Control, Institute of Systems Science, The Chinese Academy of Sciences, Beijing 100080, China. E-mail address: hfchen@iss03.iss.ac.cn
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Abstract

The Robbins–Monro algorithm with randomly varying truncations for measurements with non-additive noise is considered. Assuming that the function under observation is locally Lipschitz-continuous in its first argument and that the noise is a φ-mixing process, strong consistency of the estimate is shown. Neither growth rate restriction on the function, nor the decreasing rate of the mixing coefficients are required.

Keywords

Stochastic approximationnon-additive noiserandomly varying truncation

MSC classification

Primary: 62L20: Stochastic approximation
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 407 - 417
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Supported by the National Climbing Project of China and the National Natural Science Foundation of China.

References

Chen, H.F. (1994). Stochastic approximation and its new applications. In Proc. 1994 Hong Kong International Workshop on New Directions of Control and Manufacturing, 2–12.Google Scholar
Chen, H.F., Duncan, T.E., and Pasik-Duncan, B. On ODE approach to parameter identification and its application to adaptive control. Preprint.Google Scholar
Chen, H.F., and Zhu, Y.M. (1986). Stochastic approximation procedure with randomly varying truncations. Scientia Sinica (series A), 29, 914–926.Google Scholar
Kushner, H.J. (1981). Stochastic approximation with discontinuous dynamics and state dependent noise: w.p. l and weak convergence. J. Math. Anal. Appl. 82, 527–542.CrossRefGoogle Scholar
Kushner, H.J., and Clark, D.S. (1978). Stochastic Approximation for Constrained and Unconstrained Systems. Springer, 28–38.CrossRefGoogle Scholar
Ljung, L. (1977). Analysis of recursive stochastic algorithms. IEEE Trans. Automatic Control 22, 551–575.CrossRefGoogle Scholar
Nevelson, M.B., and Hasminskii, R.A. (1976). Stochastic approximation and recursive estimation. AMS Transl. Math. Monographs 49.CrossRefGoogle Scholar
Robbins, H., and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22, 400–407.CrossRefGoogle Scholar
Tadić, V. Stochastic optimization by a gradient algorithm with random truncations. Preprint.Google Scholar
Wang, I.J., Chong, E.K. P., and Kulkarni, S.R. (1995). On equivalence of some noise conditions for stochastic approximation algorithms. In Proceedings of the 34th CDC. New Orleans, pp. 3849–3854.Google Scholar
Yin, G., and Zhu, Y.M. (1989). Almost sure convergence of stochastic approximation algorithms with non-additive noise. Int. J. Control 49, 1361–1376.CrossRefGoogle Scholar