Rescaled fast and slow iterates of two-time-scale stochastic approximation converge weakly to decoupled Ornstein-Uhlenbeck processes, with coupling entering only through the coefficient matrices.
Convergence of probability measures
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Decoupled Functional Central Limit Theorems for Two-Time-Scale Stochastic Approximation
Rescaled fast and slow iterates of two-time-scale stochastic approximation converge weakly to decoupled Ornstein-Uhlenbeck processes, with coupling entering only through the coefficient matrices.