What happens to the Gaussian volume of a Borel set when it is translated?

Question

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, $A \subseteq \mathbb R^n$ be Borel and $c \in \mathbb R^n$. Define the translate $A_c := c + A := \{c+a \mid a \in A\} = \{x \in \mathbb R^n \mid x-c \in A\}$. Assume $A$ is not too "small", e.g $\gamma_n(A) \ge 1/2$.

Question

• What are good lower (and perhaps upper) bounds for $\gamma_n(A_c)$ ?

Observations

For a half-space $H = \{x \in \mathbb R^n \mid u^Tx \le v\}$, one has $H_c = \{x \in \mathbb R^n \mid u^Tx \le v + u^Tc\}$, another half-space with Gaussian volume $\gamma_n(H_c) = \Phi\left(\frac{v + u^Tc}{\|u\|}\right) = \Phi\left(\Phi^{-1}(\gamma_n(H)+\frac{u^Tc}{\|u\|}\right)$, where $\Phi$ is the 1D Gaussian CDF.

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Update

Let $H$ be a half-space with same Gaussian volume as $B$, i.e $\gamma_n(H)=\gamma_n(A)$.

Question: Under which minimal / reasonable conditions do we have $\gamma_n(A_c) \ge \gamma_n(H_c)$ ?

Answer 1

Let \begin{equation} g(t):=\gamma_n(tu+c+A)=P(Z\in tu+c+A)=\int_{c+A}(2\pi)^{-n/2}e^{-|z+tu|^2/2}\,dz, \end{equation} where $t\in\mathbb R$, $u$ is a unit vector in $\mathbb R^n$, $Z$ is a standard Gaussian random vector in $\mathbb R^n$, and $|\cdot|$ is the Euclidean norm. Then \begin{equation} g'(0)=-\int_{c+A}(2\pi)^{-n/2}z\cdot u\,e^{-|z|^2/2}\,dz =-p\,u\cdot E(Z|Z\in c+A), \end{equation} where $p:=P(Z\in c+A)=\gamma_n(c+A)$ and $z\cdot u$ is the dot product of $z$ and $u$.

That is, the rate of change of $\gamma_n(c+A)$ with respect to $c$ in the direction of a unit vector $u$ equals the following:

$\gamma_n(c+A)$ times the $u$-coordinate $u\cdot E(Z|Z\in c+A)$ of the center $E(Z|Z\in c+A)$ of the standard Gaussian mass over the set $c+A$.

Answer 2

It turns out that Neyman-Pearson theory helps get a nontrivial inequality.

Notations. For a p.s.d matrix $M$ of size $p$, consider the inner product on $\mathbb R^p$ defined by $\langle x,z \rangle_M := x^TMz$. This induces a norm defined by $\|x\|_M:=\sqrt{\langle x,x \rangle_M}$.

Theorem (Neyman-Pearson for translated multivaritate Gaussians). Let $\beta \in \mathbb R$, $\delta\in \mathbb R^p$ and $A$ be a Borell subset of $\mathbb R^p$. Let $X \sim \mathcal N(0,\Sigma)$ and $Y:=X+\delta$. Consider the half-space \begin{eqnarray} H=\{z \in \mathbb R^p \mid \langle\delta,z\rangle_{\Sigma^{-1}} \le \beta\}. \end{eqnarray}

• If $\mathbb P(X \in A) \ge \mathbb P(X \in H)$, then $\mathbb P(Y \in A) \ge \mathbb P(Y \in H)$.

• If $\mathbb P(X \in A) \le \mathbb P(X \in H^c)$, then $\mathbb P(Y \in A) \le \mathbb P(Y \in H^c)$.

Proof. The log of the ratio of the densities of $Y$ and $X$ is given by \begin{eqnarray*} \begin{split} \log(f_Y(z))-\log(f_X(z)) &= -\frac{1}{2}(z-\delta)^T\Sigma^{-1}(z-\delta)-\frac{1}{2}z^T\Sigma^{-1}z\\ &= \langle \delta,z \rangle_{\Sigma^{-1}}-\frac{1}{2}\|\delta\|_{\Sigma^{-1}}^2. \end{split} \end{eqnarray*} Thus $f_Y(z) \le t f_X(z)$ iff $\langle \delta,z \rangle_{\Sigma^{-1}}-\frac{1}{2}\|\delta\|_{\Sigma^{-1}}^2 \le \log(t)$. Define $t := e^{\beta-\frac{1}{2}\|\delta\|_{\Sigma^{-1}}^2}$. Then $S_t=H$, and we can apply the Neyman-Pearson Lemma (see Appendix below) to get the claimed results. $\Box$

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Appendix

The following Lemma is a modern formulation of the celebrated Neyman-Pearson Lemma.

Lemma (Neyman-Pearson 1933). Let $A$ be an event in a probability space $\mathcal Z$, and let $X$, $Y$ be random variables on $\mathcal Z$, with densities $f_X$ and $f_Y$ respectively. Finally, let $t > 0$ and define \begin{eqnarray} S_t:= \{z \in \mathcal Z \mid f_Y(z) \le tf_X(z)\}. \end{eqnarray} We have the following:

• If $\mathbb P(X \in A) \ge \mathbb P(X \in S_t)$, then $\mathbb P(Y \in A) \ge \mathbb P(Y \in S_t)$.

• If $\mathbb P(X \in A) \le \mathbb P(X \in S_t^c)$, then $\mathbb P(Y \in A) \le \mathbb P(Y \in S_t^c)$.

Proof. Suppose $\mathbb P(X \in A) \ge \mathbb P(X \in S_t)$, and let $A^c$ be the set complement of $A$ in $\mathcal Z$. One computes \begin{eqnarray*} \begin{split} &\mathbb P(Y \in A)-\mathbb P(Y \in S_t)=\int 1_A(z)f_Y(z)dz-\int_{S_t}f_Y(z)dz\\ &= \int_{S_t} 1_A(z)f_Y(z)dz+\int_{S_t^c} 1_A(z)f_Y(z)dz-\left(\int_{S_t} 1_A(z)f_Y(z)dz+\int_{S_t} 1_{A^c}(z)f_Y(z)dz \right)\\ &=\int_{S_t^c} 1_A(z)f_Y(z)dz-\int_{S_t} 1_{A^c}(z)f_Y(z)dz\\ &\ge t\left(\int_{S_t^c} 1_A(z)f_X(z)dz-\int_{S_t} 1_{A^c}(z)f_X(z)dz\right),\text{ by definition of }S_t\\ &= t(\mathbb P(X \in A)-\mathbb P(X \in S_t)) \ge 0,\text{ by assumption}. \end{split} \end{eqnarray*} Thus $\mathbb P(Y \in A) \ge \mathbb P(Y \in S_t)$. Similarly, one proves the second part of the claim with "$\ge$" replaced with "$\le$". $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\Box$