Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds

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Abstract

In this paper the Robbins-Monro (RM) algorithm with step-size an = 1 n and truncated at randomly varying bounds is considered under mild conditions imposed on the regression function. It is proved that for its a.s. convergence to the zero of a regression function the necessary and sufficient condition is ( 1 n) ∑ i=1 nξi → n→∞0 a.s. where ξi denotes the measurement error. It is also shown that the algorithm is robust with respect to the measurement error in the sense that the estimation error for the sought-for zero is bounded by a function g(ε) such that g(ε) → ε→00 if lim sup( 1 n) n→∞ ∑ i=1 nξi ε > 0. © 1988.

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Chen, H. F., Guo, L., & Gao, A. J. (1987). Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Processes and Their Applications, 27(C), 217–231. https://doi.org/10.1016/0304-4149(87)90039-1

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