The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, with integrals and integrability as in Measure Spaces and the Lebesgue Integral: Standing Notation; $\exp$ is the exponential function and $\log$ the natural logarithm. Smoothness on $\mathbb{R}^{d}$ is that of Smooth Map on a Euclidean Open Set, and for a function $f$ of class $C^{1}$ on $\mathbb{R}^{d}$, $Df(x)\in\mathbb{R}^{d}$ is its gradient at $x$; Borel is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Let $c$ be a variance vector, with the scaling map $S_{c}$ and weighted square $|x|_{c}^{2}$ ($x\in\mathbb{R}^{d}$), and let $\rho_{c}$ be the diagonal Gaussian density with variances $c$.

1. (Regularity)¶ $\rho_{c}$ is smooth, positive and Borel on $\mathbb{R}^{d}$.

2. (Gradient)¶ $D\rho_{c}(x)=-\rho_{c}(x)\,S_{c}(x)$ for every $x\in\mathbb{R}^{d}$.

3. (Normalization)¶ $\int_{\mathbb{R}^{d}}\rho_{c}\,d\lambda_{d}=1$.

4. (Second moments)¶ For every $i\in[d]$ the function $x\mapsto x_{i}^{2}\rho_{c}(x)$ is integrable with respect to $\lambda_{d}$ and $\int_{\mathbb{R}^{d}}x_{i}^{2}\rho_{c}(x)\,\lambda_{d}(dx)=c_{i}$; consequently $\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\rho_{c}(x)\,\lambda_{d}(dx)=\sum_{i=1}^{d}c_{i}$.

5. (Exponential moments of diagonal quadratic forms)¶ Let $t=(t_{1},\dots,t_{d})\in\mathbb{R}^{d}$ satisfy $2t_{i}c_{i}<1$ for every $i\in[d]$. Then the function $x\mapsto\exp\bigl(\sum_{i=1}^{d}t_{i}x_{i}^{2}\bigr)\rho_{c}(x)$ is integrable with respect to $\lambda_{d}$ and

$$\int_{\mathbb{R}^{d}}\exp\Bigl(\sum_{i=1}^{d}t_{i}x_{i}^{2}\Bigr)\rho_{c}(x)\,\lambda_{d}(dx)=\exp\Bigl(-\tfrac12\sum_{i=1}^{d}\log(1-2t_{i}c_{i})\Bigr).$$