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

H. Walk

Journal of Multivariate Analysis, 1978, vol. 8, issue 3, 430-452

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.

Keywords: Robbins-Monro process in D[0; 1] martingales in D[0; 1] strong theorems invariance principles Gaussian process Brownian motion in C[0; 1] initial value problem (search for similar items in EconPapers)
Date: 1978
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Citations: View citations in EconPapers (2)

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