  In this paper, we establish the theory of weak convergence (toward a normal
distribution) for both single-chain and population stochastic approximation
MCMC algorithms. Based on the theory, we give an explicit ratio of convergence
rates for the population SAMCMC algorithm and the single-chain SAMCMC
algorithm. Our results provide a theoretic guarantee that the population SAMCMC
algorithms are asymptotically more efficient than the single-chain SAMCMC
algorithms when the gain factor sequence decreases slower than O(1/t), where t
indexes the number of iterations. This is of interest for practical
applications.
