  We study the mixing properties of an important optimization algorithm of
machine learning: the stochastic gradient Langevin dynamics (SGLD) with a fixed
step size. The data stream is not assumed to be independent hence the SGLD is
not a Markov chain, merely a \emph{Markov chain in a random environment}, which
complicates the mathematical treatment considerably. We derive a strong law of
large numbers and a functional central limit theorem for SGLD.
