Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, with Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation in force, fix a dimension $q$, and let $g_{s}$ for $0<s\le1$, the constants $A_{0},\dots,A_{3}$ and the notation $s^{-m/2}$ be those of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails. Lebesgue measure $\lambda_{q}$ on $\mathcal{B}(\mathbb{R}^{q})$ is that of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (v), integrals with respect to it are also written $\int f(y)\,dy$, and a function on $\mathbb{R}^{q}$ is integrable when it is integrable with respect to $\lambda_{q}$; a function $\mathbb{R}^{q}\to\mathbb{R}$ is bounded as defined there, and a bounded Borel function is integrable with respect to every probability measure by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii). Let $\mu\in\mathcal{P}(\mathbb{R}^{q})$ and let $s$ satisfy $0<s\le1$.

(i) For $y\in\mathbb{R}^{q}$ the function $x\mapsto g_{s}(y-x)$ equals $x\mapsto g_{s}(x-y)$ by the evenness in The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (ii), is Borel by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, and is bounded by $A_{0}s^{-q/2}$ by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (iii); hence

$$(g_{s}*\mu)(y)=\int_{\mathbb{R}^{q}}g_{s}(y-x)\,\mu(dx)$$

is a real number, and the function $g_{s}*\mu:\mathbb{R}^{q}\to\mathbb{R}$ is called the Gaussian smoothing of $\mu$ at scale $s$. For a bounded Borel $\phi:\mathbb{R}^{q}\to\mathbb{R}$ and $x\in\mathbb{R}^{q}$ the function $y\mapsto g_{s}(x-y)\,\phi(y)$ is Borel and integrable, being dominated by a constant multiple of the function $y\mapsto g_{s}(x-y)$, which is integrable with integral $1$ by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder; we write

$$(g_{s}*\phi)(x)=\int_{\mathbb{R}^{q}}g_{s}(x-y)\,\phi(y)\,dy .$$

Then the following hold.

1. (Regularity)(ii) $g_{s}*\mu$ is of class $C^{3}$ on $\mathbb{R}^{q}$, and for all $y\in\mathbb{R}^{q}$ and $i,j,k\in[q]$,

$$\partial_{i}(g_{s}*\mu)(y)=\int_{\mathbb{R}^{q}}\partial_{i}g_{s}(y-x)\,\mu(dx),\qquad \partial_{j}\partial_{i}(g_{s}*\mu)(y)=\int_{\mathbb{R}^{q}}\partial_{j}\partial_{i}g_{s}(y-x)\,\mu(dx),$$ $$\partial_{k}\partial_{j}\partial_{i}(g_{s}*\mu)(y)=\int_{\mathbb{R}^{q}}\partial_{k}\partial_{j}\partial_{i}g_{s}(y-x)\,\mu(dx),$$

the integrands being bounded Borel functions of $x$; moreover $0\le(g_{s}*\mu)(y)\le A_{0}s^{-q/2}$, $|\partial_{i}(g_{s}*\mu)(y)|\le A_{1}s^{-(q+1)/2}$, $|\partial_{j}\partial_{i}(g_{s}*\mu)(y)|\le A_{2}s^{-(q+2)/2}$ and $|\partial_{k}\partial_{j}\partial_{i}(g_{s}*\mu)(y)|\le A_{3}s^{-(q+3)/2}$.

2. (Mass and duality)(iii) $g_{s}*\mu$ is nonnegative, Borel and integrable, with $\int_{\mathbb{R}^{q}}g_{s}*\mu\,d\lambda_{q}=1$. For every bounded Borel $\phi:\mathbb{R}^{q}\to\mathbb{R}$ with $|\phi(y)|\le M$ for all $y$, the function $g_{s}*\phi$ is Borel with $|(g_{s}*\phi)(x)|\le M$ for all $x$, the product $\phi\cdot(g_{s}*\mu)$ is integrable, and

$$\int_{\mathbb{R}^{q}}\phi\,(g_{s}*\mu)\,d\lambda_{q}=\int_{\mathbb{R}^{q}}g_{s}*\phi\,d\mu .$$

3. (Approximation)(iv) Let $\phi:\mathbb{R}^{q}\to\mathbb{R}$ be Borel and let $M$, $r$ and $\epsilon$ be positive real numbers such that $|\phi(x)|\le M$ for all $x$ and $|\phi(x)-\phi(x')|\le\epsilon$ for all $x,x'$ with $\lVert x-x'\rVert\le r$. Then for every $x\in\mathbb{R}^{q}$,

$$\bigl|(g_{s}*\phi)(x)-\phi(x)\bigr|\le\epsilon+\frac{2\,M\,q\,s}{r^{2}} .$$

4. (Pairing identity)(v) Let $\nu\in\mathcal{P}(\mathbb{R}^{q})$ and suppose $0<s\le\tfrac12$. Then the product $(g_{s}*\mu)(g_{s}*\nu)$ is integrable, the function $z\mapsto g_{2s}(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$ on $\mathbb{R}^{q+q}$ is bounded and Borel, and

$$\int_{\mathbb{R}^{q}}(g_{s}*\mu)(g_{s}*\nu)\,d\lambda_{q}=\int_{\mathbb{R}^{q+q}}g_{2s}\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\bigr)\,(\mu\boxtimes\nu)(dz),$$

with the coordinate projections and product measure of that clause.