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Martingales and the Robbins-Monro procedure in D[0, 1]

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  • Walk, H.

Abstract

The Robbins-Monro procedure for recursive estimation of a zero point of a regression function f is investigated for the case f defined on and with values in the space D[0, 1] of real-valued functions on [0, 1] that are right-continuous and have left-hand limits, endowed with Skorohod's J1-topology. There are proved an a.s. convergence result and an invariance principle where the limit process is a Gaussian Markov process with paths in the space of continuous C[0, 1]-valued functions on [0, 1]. At first the case f(x) [reverse not equivalent] x, i.e., the case of a martingale in D[0, 1], is treated and by this then the general case. An application to an initial value problem with only empirically available function values is sketched.

Suggested Citation

  • Walk, H., 1978. "Martingales and the Robbins-Monro procedure in D[0, 1]," Journal of Multivariate Analysis, Elsevier, vol. 8(3), pages 430-452, September.
    • Handle: RePEc:eee:jmvana:v:8:y:1978:i:3:p:430-452
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    Cited by:

    1. Soonhui Lee & Tito Homem-de-Mello & Anton Kleywegt, 2012. "Newsvendor-type models with decision-dependent uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(2), pages 189-221, October.
    2. Yue Wang & Joseph G. Ibrahim & Hongtu Zhu, 2020. "Partial least squares for functional joint models with applications to the Alzheimer's disease neuroimaging initiative study," Biometrics, The International Biometric Society, vol. 76(4), pages 1109-1119, December.

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