An Almost Sure Central Limit Theorem for Stochastic Approximation Algorithms

We prove an almost sure central limit theorem for some multidimensional stochastic algorithms used for the search of zeros of a function and known to satisfy a central limit theorem. The almost sure version of the central limit theorem requires either a logarithmic empirical mean (in the same way as in the case of independent identically distributed variables) or another scale, depending on the choice of the algorithm gains