Let \(\mu _m\) and \(\nu \) be probability measures on \(\mathbb R\). Assume that for every \(c\ge 0\) and every \(c\)-Lipschitz \(\psi :\mathbb R\to \mathbb R\), \(\int \psi \, d\mu _m\to \int \psi \, d\nu \). Then for each \(n\ge 0\) and every bounded continuous \(F:\mathbb R^n\to \mathbb R\), \(\int F\, d\mu _m^{\otimes n}\to \int F\, d\nu ^{\otimes n}\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). Let \(\nu _m\) be the law of the normalized sum of \(m+1\) independent uniform signs. For every \(n\ge 0\) and bounded continuous \(F:\mathbb R^n\to \mathbb R\), \(\int F\, d\nu _m^{\otimes n}\to \int F\, d\gamma _n\) as \(m\to \infty \).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For independent uniform bits \(X_{ij}\), put \(S_r(X)_i=r^{-1/2}\sum _{j=1}^r(-1)^{X_{ij}}\). For every \(n\ge 0\) and bounded continuous \(F:\mathbb R^n\to \mathbb R\), \(\mathbb E F(S_{m+1}(X))\to \int F\, d\gamma _n\) as \(m\to \infty \).