  The subject of stochastic approximation was founded by Robbins and Monro
[Ann. Math. Statist. 22 (1951) 400--407]. After five decades of continual
development, it has developed into an important area in systems control and
optimization, and it has also served as a prototype for the development of
adaptive algorithms for on-line estimation and control of stochastic systems.
Recently, it has been used in statistics with Markov chain Monte Carlo for
solving maximum likelihood estimation problems and for general simulation and
optimizations. In this paper, we first show that the trajectory averaging
estimator is asymptotically efficient for the stochastic approximation MCMC
(SAMCMC) algorithm under mild conditions, and then apply this result to the
stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer.
Statist. Assoc. 102 (2007) 305--320]. The application of the trajectory
averaging estimator to other stochastic approximation MCMC algorithms, for
example, a stochastic approximation MLE algorithm for missing data problems, is
also considered in the paper.
