Large-Time Behavior of Stochastic Approximation Processes

Asymptotic normality is proved for certain continuous stochastic approximation (SA) procedures under reasonable assumptions, along with the following more general result: under suitable coordinate and time transformations a SA process will asymptotically approach a Gaussian Markov process. As in [R. Z. Khas'minskii, Stability of Systems of Differential Equations under Random Disturbances of Their Parameters, Fizmatgiz, Moscow, 1969; M. B. Nevel'son, R. Z. Khas'minskii, Probl. Peredachi Inf., 1971, vol. 7, no. 2, pp. 58–69], the limit theorems and properties of SA procedures are deducted from theorems on the solutions of systems, stable in a certain sense, of stochastic differential equations.