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.
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Dippon, J., Walk, H. The Averaged Robbins – Monro Method for Linear Problems in a Banach Space. J Theor Probab 19, 166–189 (2006). https://doi.org/10.1007/s10959-006-0007-4
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DOI: https://doi.org/10.1007/s10959-006-0007-4