For \(v\gt 0\) and \(M\ge 0\), let \(p_v(m,z)\) be the Gaussian density with mean \(m\) and variance \(v\). There exists a nonnegative Lebesgue-integrable \(b:\mathbb R\to \mathbb R\) such that \(p_v(m,z)\le b(z)\), \(|((z-m)/v)p_v(m,z)|\le b(z)\), and \(|((z-m)^2/v^2-1/v)p_v(m,z)|\le b(z)\) whenever \(|m|\le M\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For \(c,z,h\in \mathbb R^I\), \(A_v(c,z)[h]=G_v(c,z)\sum _i((z_i-c_i)/v)h_i\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For every \(c,z\in \mathbb R^I\), the derivative at \(c\) of \(m\mapsto G_v(m,z)\) is \(A_v(c,z)\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). If \(v\gt 0\) and \(M\ge 0\), there exists a nonnegative Lebesgue-integrable \(b:\mathbb R^I\to \mathbb R\) such that \(|G_v(m,z)|\le b(z)\) and \(\| A_v(m,z)\| \le b(z)\) for all \(z\) and all \(m\) with \(|m_i|\le M\) for every \(i\), with operator norm for the maximum norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For each fixed \(v\ge 0\), \((c,z)\mapsto A_v(c,z)\) is continuous in operator norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For fixed \(v\ge 0\) and \(c\in \mathbb R^I\), \(z\mapsto A_v(c,z)\) is continuous in operator norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\) and \(c\in \mathbb R^I\), then \(z\mapsto f(z)A_v(c,z)\) is Lebesgue-integrable as a linear-functional-valued function.

Let \(I\) be finite and \(c,m\in \mathbb R^I\). If \(\| m-c\| _\infty \lt \varepsilon \) for a real \(\varepsilon \), then \(|m_i|\le \| c\| _\infty +\varepsilon \) for every \(i\in I\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the function \(m\mapsto \int _{\mathbb R^I}G_v(m,z)f(z)\, dz\) is differentiable at every \(c\), with derivative \(\int _{\mathbb R^I}f(z)A_v(c,z)\, dz\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the linear-functional-valued function \(c\mapsto \int _{\mathbb R^I}f(z)A_v(c,z)\, dz\) is continuous.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the map \(m\mapsto \int _{\mathbb R^I}G_v(m,z)f(z)\, dz\) is continuously differentiable.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), \(c,h\in \mathbb R^I\), and \(Z\sim \bigotimes _iN(c_i,v)\), then \((\int f(z)A_v(c,z)\, dz)[h]=\sum _i\mathbb E[((Z_i-c_i)/v)f(Z)]h_i\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\) and \(c\in \mathbb R^I\), then \(\| \int f(z)A_v(c,z)\, dz\| \le |I|(1+1/v)\), in operator norm for the maximum norm.