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

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 D0, 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 C0, 1]-valued functions on 0, 1]. At first the case f(x) ≡ x, i.e., the case of a martingale in D0, 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.