Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(c\in \mathbb R\), and \(f:\Omega \to \mathbb R\). Then \(\operatorname {Stab}_{c^2}(f)=\| T_cf\| _{2,P}^2\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\operatorname {Stab}_a(h)=\mathbb E_P[hT_a h]\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), an event \(A\subseteq \Omega \), and \(p\gt 0\). Then \(\| \mathbf1_A\| _{p,P}^2=\Pr _P(A)^{2/p}\). For a real function \(h\) on \(\Omega \), use \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\), and let \(q\ge 2\) and \(0\le \rho \le \lambda ^{1-2/q}/(q-1)\). For every event \(A\subseteq \Omega \), \(\operatorname {Stab}_\rho (\mathbf1_A)\le \Pr _P(A)^{2-2/q}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\operatorname {Stab}_a(h)=\mathbb E_P[hT_a h]\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere. The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \ge 0\), a coordinate \(i\), and \(f:\Omega \to \mathbb R\). Then \(\rho I_i^{(\rho )}(f)=\| T_{\sqrt\rho }L_if\| _{2,P}^2\), where \(I_i^{(\rho )}\) is stable influence and \(L_i\) is the coordinate Laplacian. For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(L_i h=h-E_{[n]\setminus \{ i\} }h\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \{ -1,1\} \), and \(S\subseteq [n]\). For conditional expectation \(E_S\) onto \(S\), \(\mathbb E_P|f-E_Sf|=\mathbb E_P(f-E_Sf)^2\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \{ -1,1\} \), a coordinate \(i\), and \(1\le p\le 2\). Then \(\mathbb E_P|L_if|^p\le I_i(f)\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(L_i h=h-E_{[n]\setminus \{ i\} }h\); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\ge 2\), \(\rho \ge 0\), \(f:\Omega \to \{ -1,1\} \), and a coordinate \(i\). If \(\| T_{\sqrt\rho }L_if\| _{2,P}\le \| L_if\| _{q/(q-1),P}\), then \(\rho I_i^{(\rho )}(f)\le I_i(f)^{2-2/q}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(L_i h=h-E_{[n]\setminus \{ i\} }h\); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\), and let \(q\ge 2\) and \(0\le \rho \le \lambda ^{1-2/q}/(q-1)\). For every \(f:\Omega \to \{ -1,1\} \) and coordinate \(i\), \(\rho I_i^{(\rho )}(f)\le I_i(f)^{2-2/q}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\). For every \(f:\Omega \to \{ -1,1\} \) and coordinate \(i\), \(\sum _{S\ni i}(\sqrt\lambda /3)^{|S|}\mathbb E_Pf_S^2\le I_i(f)^{3/2}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1/2\). If \(f:\Omega \to \{ -1,1\} \) has positive variance and \(a=I(f)/\operatorname {Var}_P(f)\), there is a coordinate \(i\) such that \((a^{-1})^2(9/\lambda )^{-a}\le I_i(f)\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I(h)=\sum _{i=0}^{n-1}I_i(h)=\sum _{S\subseteq [n]}|S|\mathbb E_Ph_S^2\); \(\operatorname {Var}_P(h)=\mathbb E_P(h-\mathbb E_Ph)^2\).

There is an absolute \(c\gt 0\) such that for every \(n\ge 2\), every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with every coordinate atom at least \(0\lt \lambda \le 1/2\), and every \(f:\Omega \to \{ -1,1\} \), some coordinate \(i\) satisfies \(c\operatorname {Var}_P(f)\log n/(n\log (1/\lambda ))\le I_i(f)\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(\operatorname {Var}_P(h)=\mathbb E_P(h-\mathbb E_Ph)^2\).

There is an absolute \(C\gt 0\) such that for every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with every coordinate atom at least \(0\lt \lambda \le 1/2\), every \(f:\Omega \to \{ -1,1\} \), and every \(0\lt \varepsilon \le 1\), there exist \(J\subseteq [n]\) and \(g:\Omega \to \{ -1,1\} \) depending only on \(J\) with \(|J|\le (1/\lambda )^{CI(f)/\varepsilon }\) and \(\Pr _P(f\ne g)\le \varepsilon \). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I(h)=\sum _{i=0}^{n-1}I_i(h)=\sum _{S\subseteq [n]}|S|\mathbb E_Ph_S^2\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere.

For every real \(\lambda \), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(\lim _{\lambda \to 0^+}C_\lambda \sqrt\lambda =1\).

For every real \(\lambda \), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(\lim _{\lambda \to (1/2)^-,\ 0\lt \lambda \lt 1/2}C_\lambda =e\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \lt 1/2\), and let \(q\gt 2\). Define \(r_q(\lambda )=\sqrt{\frac{\sinh (\log ((1-\lambda )/\lambda )/q)}{\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}}\); if \(0\le \rho \le r_q(\lambda )^2\), then \(\operatorname {Stab}_\rho (\mathbf1_A)\le \Pr _P(A)^{2-2/q}\) for every event \(A\subseteq \Omega \). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\operatorname {Stab}_a(h)=\mathbb E_P[hT_a h]\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere. The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \lt 1/2\), and let \(q\gt 2\). Define \(r_q(\lambda )=\sqrt{\frac{\sinh (\log ((1-\lambda )/\lambda )/q)}{\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}}\); if \(0\le \rho \le r_q(\lambda )^2\), then \(\rho I_i^{(\rho )}(f)\le I_i(f)^{2-2/q}\) for every \(f:\Omega \to \{ -1,1\} \) and coordinate \(i\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).