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 ease of independent identically distributed variables) or another scale, depending on the choice of the algorithm gains. (C) 1999 Academic Press.