Let $T_n$ be a strongly consistent estimator of a parameter $\theta$. Are there algorithms for accelerating the convergence of $\lbrace T_n \rbrace$ to $\theta?$ In the case of a deterministic sequence $\lbrace x_n \rbrace $ that converges to $x^{\ast},$ I could use, for example, Aitken's acceleration. Is there a similar approach to accelerating stochastic sequences?
Here is my specific situation. I am doing simulations to estimate a parameter using a consistent estimator. I increase my sample size successively as $n=100, 200, 400, 800, \ldots.$ For each sample size, I run 1000 simulations and obtain the estimate of the parameter as the mean across simulations. Is there a way to use the estimates (mean and variance) at each sample size to improve the convergence of my estimator? Is there something similar to Aitken's acceleration for improving the rate of convergence of stochastic sequences?