We study the rate of convergence of linear two-time-scale stochastic approximation methods. We consider two-time-scale linear iterations driven by i.i.d. noise, prove some results on their asymptotic covariance and establish asymptotic normality. The well-known result [Polyak, B. T. (1990). Automat. Remote Contr. 51 937–946; Ruppert, D. (1988). Technical Report 781, Cornell Univ. ] on the optimality of Polyak–Ruppert averaging techniques specialized to linear stochastic approximation is established as a consequence of the general results in this paper.
"Convergence rate of linear two-time-scale stochastic approximation." Ann. Appl. Probab. 14 (2) 796 - 819, May 2004. https://doi.org/10.1214/105051604000000116