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 $m\leq 2$.

Joint with Alex Tarter.  https://arxiv.org/abs/2202.09309