Antithetic variance and a monotone sufficient condition

For a uniform coordinate, $\mathbb{P}(U_{1j}=s)=0$ for every $s\in [0,1]$, since $\mathbb{P}(U_{1j}\le s)=s$ and the left limit of this cdf at $s$ is also $s$. Thus for $0\le t\le 1$, $\mathbb{P}(1-U_{1j}\le t)=\mathbb{P}(U_{1j}\ge 1-t)=1-\mathbb{P}(U_{1j}<1-t)=t$. By [F3], the reflected coordinates remain independent; [F4] then shows $1-U_1$ has the same product-uniform law as $U_1$. Therefore both $T(U_1)$ and $T(1-U_1)$ have law $P$. Applying [F10] to $f$ and $f^2$ identifies their means and variances as $\mu$ and ${\sigma }^2$, so $Y$ and $\tilde{Y}$ have the same law, mean $\mu$, and variance ${\sigma }^2$.

The pair value $Z_i=[f(T(U_i))+f(T(1-U_i))]\mathrm{/}2$ is a measurable function of $U_i$. Thus the $Z_i$ are IID by [F2]–[F3], each has mean $\mu$ by [F9], and their average is unbiased.

By [F5]–[F6], $\mathrm{Var}(Z_i)=[\mathrm{Var}(Y)+\mathrm{Var}(\tilde{Y})+2\mathrm{Cov}(Y,\tilde{Y})]\mathrm{/}4=[{\sigma }^2+\mathrm{Cov}(Y,\tilde{Y})]\mathrm{/}2$. For distinct $i,j$, the centered $Z_i,Z_j$ are independent, so [F7] gives zero covariance; expanding the centered square of their average therefore gives the stated variance divided by $n$.

The mean of $2n$ independent $P$-draws has variance ${\sigma }^2\mathrm{/}(2n)$ by the same centered-square expansion and [F7]. Comparing this with step 2.1 gives the iff conditions for weak and strict variance reduction.

For each coordinate choose a monotonicity direction for $g$: define $S_j(v)=v_j$ in a nondecreasing coordinate and $S_j(v)=1-v_j$ in a nonincreasing coordinate; a constant coordinate may be assigned either direction. Then $S$ is a coordinatewise reflection map with $S(S(v))=v$ and $S(1-v)=1-S(v)$. By [F3] and [F4], $V=S(U_1)$ again has independent uniform coordinates, and $\varphi =g\circ S$ is coordinatewise nondecreasing on $[0,1{]}^d$. Define the coordinatewise clamp $C:{\mathbb{R}}^d\to [0,1{]}^d$ by $C_j(x)=\min \{1,\max \{0,x_j\}\}$. It is continuous and nondecreasing in every coordinate, and $C(1-v)=1-C(v)$. Hence $\bar{\varphi }:=\varphi \circ C$ is a Borel coordinatewise nondecreasing function on ${\mathbb{R}}^d$, while $v\mapsto \bar{\varphi }(1-v)$ is Borel and coordinatewise nonincreasing. On the cube these extensions equal $\varphi (v)$ and $\varphi (1-v)$, respectively; the identities for $S$ give $g(U_1)=\varphi (V)$ and $g(1-U_1)=\varphi (1-V)$. By step 1.1, these functions are square-integrable, so [F8] applies to $\bar{\varphi }(V)$ and $\bar{\varphi }(1-V)$ and gives $\mathrm{Cov}(Y,\tilde{Y})\le 0$. In the stated ordered-subset case, $u\le v$ coordinatewise implies $T(u)\le T(v)$ and then $f(T(u))\le f(T(v))$, which makes $g$ nondecreasing and is the special case $S(v)=v$.