Antithetic randomized sampling

Let $d,n\ge 1$, let $\mathsf{X}$ be a measurable space, let $P$ be a probability law on $\mathsf{X}$, and let $f:\mathsf{X}\to \mathbb{R}$ be measurable. On a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, let $U_i=(U_{i1},\dots ,U_{id})$, $i=1,\dots ,n$, be IID random vectors whose coordinates are mutually independent and each have the uniform law on $[0,1]$, meaning $\mathbb{P}(U_{ij}\le t)=t$ for $0\le t\le 1$. Let $T:[0,1{]}^d\to \mathsf{X}$ be measurable and satisfy $T(U_1)\sim P$. Define the reflected input $1-U_i=(1-U_{i1},\dots ,1-U_{id})$ and the antithetic estimator by

${\widehat{\mu }}_{\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}}=\frac{1}{n}{\sum }_{i=1}^n\frac{f(T(U_i))+f(T(1-U_i))}{2}\mathrm{.}$

The two outputs in each pair may be dependent; pairs with different indices are independent. The variance comparison in the theorem uses $2n$ function evaluations and compares with an ordinary mean of $2n$ independent $P$-draws. IID and mutual independence have the meanings of Identical distribution and IID families, and variance/covariance use Moments, variance, and covariance on a probability space.