Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(0\le c\le 1\), and \(f:\Omega \to \mathbb R\). Then \(\| T_cf\| _{2,P}\le \| f\| _{2,P}\). 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 \(p\gt 0\). 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 \(\Omega =\prod _i\Omega _i\), \(c\ge 1\), \(k\in \mathbb N\), and \(f:\Omega \to \mathbb R\) of degree at most \(k\). Then \(\| T_cf\| _{2,P}\le c^k\| f\| _{2,P}\). 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 \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\). 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 \(\lambda \), where \(0\lt \lambda \le 1\), and let \(q\gt 2\). For \(k\in \mathbb N\) and \(f:\Omega \to \mathbb R\) of degree at most \(k\), \(\| f\| _{q,P}\le (\sqrt{q-1}\lambda ^{1/q-1/2})^k\| f\| _{2,P}\). 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); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

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 \(0\lt \lambda \lt 1/2\). For \(f:\Omega \to \mathbb R\) of degree at most \(k\in \mathbb N\), \(\| f\| _{2,P}\le C_\lambda ^k\| f\| _{1,P}\), where \(C_\lambda =((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))}\). 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); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

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 \(\lambda \), where \(0\lt \lambda \le 1/2\). Define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\) For \(f:\Omega \to \mathbb R\) of degree at most \(k\in \mathbb N\), \(\| f\| _{2,P}\le C_\lambda ^k\| f\| _{1,P}\). 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); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

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

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 \(f:\Omega \to \mathbb R\) has degree at most \(k\in \mathbb N\) and positive variance, then \(\tfrac 14(e^2/(2\lambda ))^{-k}\le \Pr _P(\mathbb E_Pf\lt f)\) and \((15/\lambda )^{-k}\le \tfrac 14(e^2/(2\lambda ))^{-k}\). 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); \(\operatorname {Var}_P(h)=\mathbb E_P(h-\mathbb E_Ph)^2\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere; degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

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 \(0\lt \lambda \le 1\). If \(k\gt 0\) is an integer, \(f:\Omega \to \mathbb R\) has degree at most \(k\), \(\| f\| _{2,P}\gt 0\), and \(t\ge (\sqrt{2e/\lambda })^k\), then \(\Pr _P(|f|\ge t\| f\| _{2,P})\le \lambda ^k\exp (-k\lambda t^{2/k}/(2e))\). 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); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere; degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(0\le c\le 1\), and \(f:\Omega \to \mathbb R\). Then \(\| T_cf\| _{2,P}\le \| f\| _{2,P}\). 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 \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).