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The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

Author

Listed:
  • Jürgen Dippon

    (Universität Stuttgart)

  • Harro Walk

    (Universität Stuttgart)

Abstract

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.

Suggested Citation

  • Jürgen Dippon & Harro Walk, 2006. "The Averaged Robbins – Monro Method for Linear Problems in a Banach Space," Journal of Theoretical Probability, Springer, vol. 19(1), pages 166-189, January.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:1:d:10.1007_s10959-006-0007-4
    DOI: 10.1007/s10959-006-0007-4
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    References listed on IDEAS

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    1. Walk H., 1988. "Limit Behaviour Of Stochastic Approximation Processes," Statistics & Risk Modeling, De Gruyter, vol. 6(1-2), pages 109-128, February.
    2. Kuelbs, J., 1973. "The invariance principle for Banach space valued random variables," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 161-172, June.
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