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On the almost sure convergence of a general stochastic approximation procedure

Published online by Cambridge University Press:  17 April 2009

S. N. Evans  and
N. C. Weber
S. N. Evans
Affiliation:
Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge, England.
N. C. Weber
Affiliation:
Department of Mathematical Statistics, University of Sydney, New South Wales 2006, Australia.
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Abstract

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A set of conditions for the almost sure convergence of a stochastic iterative procedure is given. The conditions are framed in terms of the behaviour of the random adjustment made the n-th step rather than in terms of some underlying regression model.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 3 , December 1986 , pp. 335 - 342
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Blum, J.R., “Approximation methods that converge with probability one” Ann. Math. Statist. 25 (1954), 382–386.Google Scholar
[2]Burkholder, D.L., “On a class of stochastic approximation procedures”, Ann. Math. Statist. 27 (1956), 1044–1059.Google Scholar
[3]Dvoretzky, A., “On stochastic approximation”, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. I (1956), 39–55.Google Scholar
[4]Friedman, S., “On stochastic approximation”, Ann. Math. Statist., 34 (1963), 343–346.Google Scholar
[5]Goodsell, C.A. and Hanson, D.L., “Almost sure convergence for the Robbins-Monro process”, Ann. Probab. 4 (1976), 890–901.CrossRefGoogle Scholar
[6]Hall, P. and Heyde, C.C., Martingale Limit Theory and its Application (Academic Press, New York, 1980).Google Scholar
[7]Hotelling, H., “Experimental determination of the maximum of a function”, Ann. Math. Statist. 12 (1941) 20–46.Google Scholar
[8]Kiefer, J. and Wolfowitz, J., “Stochastic estimation of the maximum of a regression function”, Ann. Math. Statist. 23 (1952), 462–466.Google Scholar
[9]Ljung, L., “Strong convergence of a stochastic approximation algorithm”, Ann. Statist. 6 (1978), 680–696.Google Scholar
[10]Robbins, H. and Monro, S., “A stochastic approximation method”, Ann. Math. Statist. 22 (1951), 400–407.Google Scholar
[11]Ruppert, D., “Almost sure approximations to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise”, Ann. Probab. 10 (1982), 178–187.Google Scholar
[12]Solo, V., “Stochastic approximation with dependent noise”, Stoch. Processes and their Applications, 13 (1982), 157–170.Google Scholar
[13]VonBahr, B. and Esseen, C.G., “Inequalities for the rth absolute moment of a sum of random variables, 1≤r≤2”, Ann. Math. Statist. 36 (1965), 299–303.Google Scholar
[14]Wasan, M.T., Stochastic approximation, (Cambridge university press, 1969).Google Scholar