Chen, Han-Fu; Guo, Lei; Gao, Ai-Jun Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. (English) Zbl 0632.62082 Stochastic Processes Appl. 27, No. 2, 217-231 (1988). In this paper the Robbins-Monro algorithm with step-size \(a_ n=1/n\) 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 \[ (1/n)\sum^{n}_{i=1}\xi_ i\to_{n\to \infty}0\quad a.s. \] where \(\xi_ i\) 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(\(\epsilon)\) such that \[ g(\epsilon)\to_{\epsilon \to 0}0\text{ if } \limsup_{n\to \infty}(1/n)\| \sum^{n}_{i=1}\xi_ i\| \equiv \epsilon >0. \] Cited in 2 ReviewsCited in 40 Documents MSC: 62L20 Stochastic approximation Keywords:randomly varying truncation; robustness; Robbins-Monro algorithm; necessary and sufficient condition; measurement error; estimation error PDFBibTeX XMLCite \textit{H.-F. Chen} et al., Stochastic Processes Appl. 27, No. 2, 217--231 (1988; Zbl 0632.62082) Full Text: DOI References: [1] Robbins, H.; Monro, S., A stochastic approximation method, Ann. Math. Statist., 22, 1, 400-407 (1951) · Zbl 0054.05901 [2] Nevel’son, M. B.; Has’minskii, R. Z., Stochastic Approximation and Recursive Estimation (1973), American Mathematical Society · Zbl 0355.62075 [3] Ljung, L., Analysis of recursive stochastic algorithms, IEEE Trans., AC-22, 4, 551-575 (1977) · Zbl 0362.93031 [4] Ljung, L., Strong convergence of a stochastic approximation algorithm, Ann. Statist., 6, 3, 680-696 (1978) · Zbl 0402.62060 [5] Kushner, H. J.; Clark, D. S., Stochastic Approximation Methods for Constrained and Unconstrained Systems (1978), Springer: Springer New York · Zbl 0381.60004 [6] Solo, V., Stochastic approximation with dependent noise, Stochastic Process. Appl., 13, 157-170 (1982) · Zbl 0496.62071 [7] Chen, H. F., Stochastic approximation with ARMA measurement errors, J. of Systems Science and Mathematical Sciences, 2, 3, 227-239 (1982) [8] H.F. Chen and Y.M. Zhu, Stochastic approximation procedure with randomly varying truncations, Scientia Sinica (Series A), 29(9),914-926; H.F. Chen and Y.M. Zhu, Stochastic approximation procedure with randomly varying truncations, Scientia Sinica (Series A), 29(9),914-926 · Zbl 0613.62107 [9] Chen, H. F., Stochastic approximation with randomly varying truncations for the optimization problem, Acta Mathematicae Applicatae Sinica, 10, 1, 58-67 (1987) · Zbl 0617.62089 [10] Deng, S. H., Constrained stochastic approximation algorithm with randomly varying truncations, Acta Mathematicae Applicatae Sinica (1986) [11] Chen, H. F.; Guo, L., Simultaneous estimation of both the zero of regression function and parameters in noise, J. of Systems Science and Mathematical Sciences, 7, 2, 119-128 (1987) · Zbl 0627.62085 [12] Deng, S. H., A combined algorithm for identification and approximation, Acta Mathematicae Applicatae Sinica (1987), to appear [13] Clark, D. S., Necessary and sufficient conditions for the Robbins-Monro method, Stochastic Process. Appl., 17, 359-367 (1984) · Zbl 0537.62068 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.