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 $0\lt \varepsilon \lt |\rho |$ of being noninteractively simulated from k correlated Gaussian samples has run time bounded by $(5/\varepsilon )^{m(\log (\varepsilon /2) / \log |\rho |)^{m^{2}}}$ , 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 $0\lt \varepsilon \lt |\rho |$ 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.