[...]... degener- xii STOCHASTIC APPROXIMATION AND ITS APPLICATIONS ate; asymptotic normality of and asymptotic efficiency by averaging method Starting from Chapter 4 the general theory developed so far is applied to different fields Chapter 4 deals with optimization by using stochastic approximation methods Convergence and convergence rates of the Kiefer-Wolfowitz (KW) algorithm with expanding truncations and randomized... that are right continuous and have left-hand limits, endowed with the Skorohod topology Convergence of to a continuous function in the Skorohod topology is equivalent to the uniform convergence on any bounded interval Let and be probability measures determined by stochastic processes and respectively on with induced by the Skorohod topology 22 STOCHASTIC APPROXIMATION AND ITS APPLICATIONS If for any... theorems for martingales and martingale difference sequences is provided in detail in Appendix B The book is written for students, engineers and researchers working in the areas of systems and control, communication and signal processing, optimization and operation research, and mathematical statistics HAN-FU CHEN Acknowledgments The support of the National Key Project of China and the National Natural... for small and large By the mean theorem there exists a vector with components located in-between the corresponding components of and such that Notice that by (1.4.2) the left-hand side of (1.4.6) is of for all sufficiently large since is bounded From this it follows that i) for small enough and large enough and hence since follows that as and ii) the last term in (1.4.8) is of From (1.4.7) and (1.4.8)... it then Since the interval does not contain the origin Noticing that we find and that there is such that STOCHASTIC APPROXIMATION AND ITS APPLICATIONS 20 for sufficientlysmall is such that for all large and all large enough and small enough from (1.4.9) we have Then by A1.4.2 there As mentioned above for sufficiently large and small enough where denotes a magnitude tending to zero as Taking (1.4.4)... in A1.2.2 Expanding to the Taylor series, we obtain where and denote the gradient and Hessian of respectively, is a vector with components located in-between the corresponding components of and and denotes the constant such that (by A1.2.2) Noticing that is and taking conditional expectation for (1.2.6), by (1.2.4) we derive Since Denoting by (A1.2.1), we have ROBBINS-MONRO ALGORITHM and noticing follows... conditions for convergence Stochastic approximation algorithms with expanding truncations with TS method are also applied to adaptive filters with and without constraints As a result, conditions required for convergence have been considerably improved in comparison with the existing results Finally, the expanding truncation technique and TS method are applied to the asynchronous stochastic approximation In the... function such that as and 12 STOCHASTIC APPROXIMATION AND ITS APPLICATIONS In order to describe conditions on noise, we introduce an integervalued function for any and any integer For define Noticing that tends to zero, for any fixed diverges to infinity as In fact, counts the number of iterations starting from time as long as the sum of step sizes does not exceed The integer-valued function will be... them to a root-seeking problem, the structural errors are unavoidable, and they are state-dependent The expanding truncation technique equipped with TS method appears a powerful tool in dealing with various parameter estimation problems: it not only has succeeded in essentially weakening conditions for convergence of the general stochastic approximation algorithm but also has made stochastic approximation. .. value is far from the true root and hence will never converge to The algorithm (1.2.2) is now called Robbins-Monro (RM) algorithm STOCHASTIC APPROXIMATION AND ITS APPLICATIONS 6 The classical approach to convergence analysis of SA algorithms is based on the probabilistic analysis for trajectories We now present a typical convergence theorem by this approach Related concept and results from probability . are listed at the end of this volume. Stochastic Approximation and Its Applications by Han-Fu Chen Institute of Systems Science, Academy of Mathematics and System Science, Chinese Academy of. Normality v ix xv 1 2 4 10 16 21 23 25 26 28 41 45 49 57 67 82 93 95 96 103 113 vi STOCHASTIC APPROXIMATION AND ITS APPLICATIONS 3.4 3.5 Asymptotic Efficiency Notes and References 4. OPTIMIZATION BY STOCHASTIC APPROXIMATION 4.1 4.2 4.3 4.4 4.5 4.6 Kiefer-Wolfowitz.