In this paper we consider a class of stochastic approximation procedures that arises in linear filtering and regression theory. Our main result asserts that the stochastic approximation process satisfies an almost sure invariance principle (with a certain rate of convergence) if the partial sums of the errors do.
"An almost sure invariance principle for stochastic approximation procedures in linear filtering theory." Ann. Appl. Probab. 7 (2) 444 - 459, May 1997. https://doi.org/10.1214/aoap/1034625339