Gaussian Smoothing of a Finitely Supported Probability Mass Function: Density Bounds and Control of the Directional Score Integral by the Move Information

Let $m\ge1$ be a natural number, let $\eta$ be a real number with $0<\eta\le1$, and let $\varphi_\eta$, $c_\eta$, $Z_b$, $\kappa_b=\lVert b\rVert^{2}/\eta$ and $R_b$ (for $b\in\mathbb{R}^m$) be as in The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, whose conventions for Euclidean space $\mathbb{R}^m$ (including sums, scalar multiples and differences of points, which make $\mathbb{R}^m$ a real vector space), the Euclidean norm, Lebesgue measure $\lambda_m$, measurability, sequential continuity, integrals, integrability, partial derivatives and the notation $t^{1/2}$ for the nonnegative square root are in force. Sums over a nonempty finite index set are those of Sum over a Finite Index Set, with the convention $\sum_{x\in\emptyset}=0$ of Sum of a Nonnegative Function over an Arbitrary Set.

Let $p$ be a discrete probability mass function on $\mathbb{R}^m$ whose support $\mathsf{S}=\{x\in\mathbb{R}^m:p(x)>0\}$ is a finite set (it is nonempty, since the sum of $p$ over $\mathbb{R}^m$ is $1$). Let $n\ge1$ be a natural number, let $a=(a_1,\dots,a_n)$ be moves in $\mathbb{R}^m$ and $w=(w_1,\dots,w_n)\in\mathbb{R}^n$ weights, let $\rho_w=\rho_{p,a,w}$ be the move score, and let $\mathsf{J}(p;a,w)$ be the move information (not to be confused with the matrix $J(q)$ of claim 2). Put $u=\sum_{j=1}^{n}w_ja_j\in\mathbb{R}^m$, $\kappa_j=\kappa_{a_j}$, and define the exit mass of the $j$th move by

Define the smoothed density $q:\mathbb{R}^m\to\mathbb{R}$ by

1. (Smoothed density) $q$ is sequentially continuous and measurable, $0<q(\theta)\le c_\eta$ for every $\theta$, and $\int_{\mathbb{R}^m}q\,d\lambda_m=1$. For every $i\in\{1,\dots,m\}$ the partial derivative of $q$ with respect to the $i$th variable exists at every $\theta\in\mathbb{R}^m$,

$\partial_iq$ is sequentially continuous with $|\partial_iq(\theta)|\le c_\eta(2\eta)^{-1/2}$ for every $\theta$, and

2. (The matrix $J(q)$) For $i,k\in\{1,\dots,m\}$ the function $\partial_iq\,\partial_kq/q$ is integrable, so that

and for every $v=(v_1,\dots,v_m)\in\mathbb{R}^m$, writing $\partial_vq=\sum_{i=1}^{m}v_i\partial_iq$,

3. (Directional score integral bound) With $\partial_uq=\sum_{i=1}^{m}u_i\partial_iq$ for $u=(u_1,\dots,u_m)$ as above,

where $\mathsf{J}(p;a,w)$ is finite, being a finite sum, and $\exp$ is the exponential function.