Dimension-Free Noninteractive Simulation from Gaussian Sources

Let $X$ and $Y$ be two real-valued random variables. Let $(X{1},Y{1}),(X{2},Y{2}),\ldots$ be independent identically distributed copies of $(X,Y)$. Suppose there are two players A and B. Player A has access to $X{1},X{2},\ldots$ and player B has access to $Y{1},Y{2},\ldots$. Without communication, what joint probability distributions can players A and B jointly simulate? That is, if $k,m$ are fixed positive integers, what probability distributions on ${1,\ldots,m}^{2}$ are equal to the distribution of $(f(X{1},\ldots,X{k}),\,g(Y{1},\ldots,Y{k}))$ for some $f,g\colon\mathbb{R}^{{k}\to{1,\ldots,m}$?} When $X$ and $Y$ are standard Gaussians with fixed correlation $\rho\in(-1,1)$, we show that the set of probability distributions that can be noninteractively simulated from $k$ Gaussian samples is the same for any $k\geq m^{2}$. Previously, it was not even known if this number of samples $m^{2}$ would be finite or not, except when $m\leq 2$. Consequently, a straightforward brute-force search deciding whether or not a probability distribution on ${1,\ldots,m}^{2}$ is within distance $0<\epsilon<|\rho|$ of being noninteractively simulated from $k$ correlated Gaussian samples has run time bounded by $(5/\epsilon)^{{m(\log(\epsilon/2)} / \log|\rho|)^{m{2}}}$, improving a bound of Ghazi, Kamath and Raghavendra. A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on ${1,\ldots,m}^{2}$ is within distance $0<\epsilon<|\rho|$ of being noninteractively simulated from $k$ samples of a given finite discrete distribution $(X,Y)$ in run time that does not depend on $k$, with constants that again improve a bound of Ghazi, Kamath and Raghavendra.