It is shown in this paper that the sample paths of a Robbins-Monro process with harmonic coefficients may be approximated by weighted sums of independent, identically distributed random variables. A law of iterated logarithm and a weak invariance principle follow from this result.
"Almost Sure Approximation of the Robbins-Monro Process by Sums of Independent Random Variables." Ann. Probab. 5 (6) 954 - 965, December, 1977. https://doi.org/10.1214/aop/1176995663