  Estimating the mixing density of a mixture distribution remains an
interesting problem in statistics literature. Using a stochastic approximation
method, Newton and Zhang (1999) introduced a fast recursive algorithm for
estimating the mixing density of a mixture. Under suitably chosen weights the
stochastic approximation estimator converges to the true solution. In Tokdar
et. al. (2009) the consistency of this recursive estimation method was
established. However, the proof of consistency of the resulting estimator used
independence among observations as an assumption. Here, we extend the
investigation of performance of Newton's algorithm to several dependent
scenarios. We prove that the original algorithm under certain conditions
remains consistent even when the observations are arising from a weakly
dependent stationary process with the target mixture as the marginal density.
We show consistency under a decay condition on the dependence among
observations when the dependence is characterized by a quantity similar to
mutual information between the observations.
