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  • Bulletin of the Australian Mathematical Society, Volume 34, Issue 3
  • December 1986, pp. 335-342

On the almost sure convergence of a general stochastic approximation procedure

  • S. N. Evans (a1) and N. C. Weber (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700010236
  • Published online: 01 April 2009
Abstract

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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]J.R. Blum , “Approximation methods that converge with probability oneAnn. Math. Statist. 25 (1954), 382386.

[2]D.L. Burkholder , “On a class of stochastic approximation procedures”, Ann. Math. Statist. 27 (1956), 10441059.

[4]S. Friedman , “On stochastic approximation”, Ann. Math. Statist., 34 (1963), 343346.

[5]C.A. Goodsell and D.L. Hanson , “Almost sure convergence for the Robbins-Monro process”, Ann. Probab. 4 (1976), 890901.

[7]H. Hotelling , “Experimental determination of the maximum of a function”, Ann. Math. Statist. 12 (1941) 2046.

[8]J. Kiefer and J. Wolfowitz , “Stochastic estimation of the maximum of a regression function”, Ann. Math. Statist. 23 (1952), 462466.

[9]L. Ljung , “Strong convergence of a stochastic approximation algorithm”, Ann. Statist. 6 (1978), 680696.

[10]H. Robbins and S. Monro , “A stochastic approximation method”, Ann. Math. Statist. 22 (1951), 400407.

[11]D. Ruppert , “Almost sure approximations to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise”, Ann. Probab. 10 (1982), 178187.

[12]V. Solo , “Stochastic approximation with dependent noise”, Stoch. Processes and their Applications, 13 (1982), 157170.

[13]B. VonBahr and C.G. Esseen , “Inequalities for the rth absolute moment of a sum of random variables, 1≤r≤2”, Ann. Math. Statist. 36 (1965), 299303.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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