Convergence of antithetic variable in Monte Carlo

Question

Monte Carlo slow convergence is due to the variance term $\frac{S_n}{\sqrt{n}}$. So there are a few method to reduce the variance of the estimator to gain convergence speed.

I'm asking about Antithetic Variable method, using two variables of negative correlation. In most of books, tutorial I've seen, they only talk about the method, and proof of variance reduction. But they never talk about the convergence proof.

Monte Carlo is based on the Law of Large Number, or Central Limit Theorem, where sequence of sample are assumed to be iid. Now if we introduce the correlation, we can not apply those two theorem, so how can we use the estimator when we cannot prove the convergence?

Answer 1

The proof of convergence is exactly the same, being based on the CLT for the two-dimensional sequence $(X_n,Y_n)$ even when $X_n$ and $Y_n$ are correlated. The estimator $$\dfrac{1}{2N}\,\sum_{i=1}^N \{h(X_i)+h(Y_i)\}$$ can be rewritten as $$\dfrac{1}{N}\,\sum_{i=1}^N H(X_i,Y_i)$$ which is asymptotically Gaussian with mean $\mathbb{E}[H(X,Y)]$ and variance $N^{-1}\text{var}\{H(X,Y)\}$. The correlation between $X$ and $Y$ does not impact this result, except quantitatively when considering the value of $\text{var}\{H(X,Y)\}$.

Note that your comment

there are a few method to reduce the variance of the estimator to gain convergence speed.

does not mean that the speed $O(1/\sqrt{n})$ changes when using antithetic variables.