On the Behavior of Moments of Arbitrary Order of Multivariate Stochastic Approximation Processes

Papers [J. Komlós and P. Révész, Stud. Sci. Math. Hung., 1973, no. 8, pp. 329–340; M. B. Nevel'son, Avtomat. Telemekh., 1973, no. 2, pp. 83–92] proposed modification of the univariate or one-dimensional Robbins–Monro (RM) procedure, which possess bounded moments of some particular order under proper normalization. It was assumed in these studies that the modulus of the regression function is bounded from below for all sufficiently large values of the modulus of the argument. This condition was subsequently removed in [M. B. Nevel'son, Properties of moments of stochastic approximation processes, Teor. Veroyatn. Primen., 1984]. In this paper, which generalizes the results of [Nevel'son,Teor. Veroyatn. Primen., 1984], we propose a method of investigating the moments of arbitrary order of a truncated KM procedure, that enables us also to consider vector-valued regression functions that decrease as the modulus of the argument increases.