Poincaré inequality in probability theory

A probability distribution $\mu$ has Poincaré constant $C_P$ when every sufficiently regular $f$ satisfies

For standard Gaussian measure $\gamma$, the sharp Poincaré constant is one:

If each probability measure ${\mu }_i$ has Poincaré constant $C_i$, then the product measure ${⨂}_i{\mu }_i$ has Poincaré constant at most ${\max }_i{C}_i$. This follows by iterating the law of total variance and applying Jensen inequality to derivatives of conditional expectations.

If $\mu$ has Poincaré constant $C$ and $\varphi$ is $L$-Lipschitz, then the pushforward measure ${\varphi }_{\ast }\mu$ has Poincaré constant at most $C{L}^2$. Apply the original inequality to $g\circ \varphi$ and use the chain rule.

For the uniform distribution on $[0,1]$,The constant $1/{\pi }^2$ is sharp and is the mean-zero Poincare-Wirtinger inequality on the interval.

For standard Gaussian measure $\gamma$,

For standard one-dimensional Gaussian measure, $\int e^{sx}d\gamma =e^{s^2/2}$. Thus ${\mathrm{Ent}}_{\gamma }(e^{sx})=(s^2/2)e^{s^2/2}$, while $\int |(e^{sx/2}{)}^{\prime }{|}^{2}d\gamma =(s^2/4)e^{s^2/2}$. Every nonzero $s$ therefore attains ratio two, proving optimality of the Gaussian constant.

For bounded smooth positive $h$ bounded away from zero, differentiation and Gaussian integration by parts give the displayed identity. The Ornstein-Uhlenbeck gradient commutation identity and weighted Cauchy-Schwarz inequality bound its right side by $e^{-2t}\int |h^{\prime }{|}^2/hd\gamma$. Integrating and setting $h=f^2$ proves the Gaussian logarithmic Sobolev inequality. Positive clipped approximation extends it to functions of finite Gaussian Dirichlet energy.

For independent random variables supported on $[0,1]$ and a differentiable convex function $f$,