There exists a universal real constant \(c\ge 0\) such that, for every \(n\ge 0\) and family of independent measurable real random variables \(X_1,\ldots ,X_n\) on any probability space, if each \(|X_i|^3\) is integrable, \(\mathbb E X_i=0\), and \(\sum _i\mathbb E X_i^2=1\), then \(D_K(\mathcal L(\sum _iX_i),\gamma )\le c\sum _i\mathbb E|X_i|^3\). Here \(\gamma \) is standard Gaussian measure and \(D_K(\mu ,\nu )=\sup _{t\in \mathbb R}|\mu (({-\infty },t])-\nu (({-\infty },t])|\) for probability measures.

There exists a universal real constant \(C\ge 0\) such that for every \(n\ge 0\), real coefficients \(a_1,\ldots ,a_n\), and real \(\varepsilon \), if \(\sum _i a_i^2=1\) and \(|a_i|\le \varepsilon \) for all \(i\), then \(|\mathbb E|\sum _i a_i\xi _i|-\sqrt{2/\pi }|\le C\varepsilon \), where the \(\xi _i\) are independent uniform signs.

Let \(G,H\) be independent standard real Gaussians and let \(-1\le \rho \le 1\). Then \(\Pr [G\le 0,\ \rho G+\sqrt{1-\rho ^2}\, H\le 0]=\tfrac 12-\arccos (\rho )/(2\pi )\).

There exists a universal real constant \(C\ge 0\) with the following property. For every \(n\ge 0\), real coefficients \(a_i\), and real \(\varepsilon ,\rho \) with \(\sum _i a_i^2=1\), \(|a_i|\le \varepsilon \) for every \(i\), \(|\rho |\lt 1\), and \(\sum _i a_i(-1)^{x_i}\ne 0\) for every \(x\in \{ 0,1\} ^n\), let \(f(x)\) equal \(1\) when \(\sum _i a_i(-1)^{x_i}\ge 0\) and \(-1\) otherwise. Then \(|\operatorname {Stab}_\rho (f)-(2/\pi )\arcsin \rho |\le C\sqrt{\max (\varepsilon ,0)}/(1-\rho )\). For a real function \(h\) on a uniform bit cube, put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\), \(\widehat h(S)=\mathbb E[h\chi _S]\), and \(\operatorname {Stab}_\rho (h)=\sum _S\rho ^{|S|}\widehat h(S)^2\).

There exists a universal real constant \(C\ge 0\) such that for every \(0\le \rho \lt 1\), the majority function \(M_k\) on \(2k+1\) uniform bits, equal to \(1\) when \(\sum _i(-1)^{x_i}\gt 0\) and \(-1\) otherwise, satisfies \(\operatorname {Stab}_\rho (M_{k+1})\le \operatorname {Stab}_\rho (M_k)\) for all \(k\ge 0\), and \(\operatorname {Stab}_\rho (M_k)\to (2/\pi )\arcsin \rho \) as \(k\to \infty \). For every integer \(k\ge 0\), \((2/\pi )\arcsin \rho \le \operatorname {Stab}_\rho (M_k)\le (2/\pi )\arcsin \rho +C/(\sqrt{1-\rho ^2}\sqrt{2k+1})\). For a real function \(h\) on a uniform bit cube, put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\), \(\widehat h(S)=\mathbb E[h\chi _S]\), and \(\operatorname {Stab}_\rho (h)=\sum _S\rho ^{|S|}\widehat h(S)^2\).

For every \(0\lt \rho \lt 1\) and \(\varepsilon \gt 0\), there exists \(\tau \gt 0\) such that for all \(n\ge 0\) and \(f\colon \{ 0,1\} ^n\to [-1,1]\) with uniform mean zero and \(I_i(f)\le \tau \) for every \(i\), one has \(\operatorname {Stab}_\rho (f)\le (2/\pi )\arcsin \rho +\varepsilon \). Here \(I_i(f)=\mathbb E[(f(x)-f(x^{\{ i\} }))^2]/4\), where \(x^{\{ i\} }\) flips bit \(i\). For a real function \(h\) on a uniform bit cube, put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\), \(\widehat h(S)=\mathbb E[h\chi _S]\), and \(\operatorname {Stab}_\rho (h)=\sum _S\rho ^{|S|}\widehat h(S)^2\).