The Diagonal Gaussian Density on Euclidean Space and Its Notation

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, $\exp$ is the exponential function, $\log$ the natural logarithm, defined on the positive reals, and $\sqrt{s}$ the nonnegative square root of $s\ge0$. Let $\kappa$ be the total mass written $c$ in claim 2 of The Gaussian Weight Defines a Probability Distribution, the integral of $s\mapsto\exp(-s^{2}/2)$ against Lebesgue measure on $\mathbb{R}$; it is a positive real number by that claim.

1. (Variances)¶ A variance vector is a $c=(c_{1},\dots,c_{d})\in\mathbb{R}^{d}$ with $c_{i}>0$ for every $i\in[d]$. For such $c$, $c_{\min}$ and $c_{\max}$ denote the least and the greatest of the real numbers $c_{1},\dots,c_{d}$. They exist: $1\le d$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data, and Greatest Element of a Finite Family in a Totally Ordered Set applied to the $d$-tuple $c$ gives an index $j$ with $c_{k}\le c_{j}$ for every $k\in[d]$, and, applied with the reverse order $\ge$ of $\mathbb{R}$ (again a total order, since reflexivity, antisymmetry, transitivity and totality are unchanged when the two arguments are interchanged), an index $j'$ with $c_{j'}\le c_{k}$ for every $k\in[d]$; so $c_{\max}=c_{j}$ and $c_{\min}=c_{j'}$, and both are positive.

2. (The scaling map and the weighted square)¶ For a variance vector $c$, $S_{c}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is the map $S_{c}(x)=(x_{1}/c_{1},\dots,x_{d}/c_{d})$, and the weighted square of $x\in\mathbb{R}^{d}$ is the real number $|x|_{c}^{2}=x\cdot S_{c}(x)=\sum_{i=1}^{d}x_{i}^{2}/c_{i}$, the second equality being the definition of the dot product; the symbol $|x|_{c}^{2}$ is used only as a whole, and no quantity $|x|_{c}$ is defined.

3. (The diagonal Gaussian density)¶ For a variance vector $c$, $Z_{c}=\sum_{i=1}^{d}\log(\kappa\sqrt{c_{i}})$, the logarithm being applied to the positive real numbers $\kappa\sqrt{c_{i}}$, and the diagonal Gaussian density with variances $c$ is the function

$$\rho_{c}:\mathbb{R}^{d}\to\mathbb{R},\qquad\rho_{c}(x)=\exp\Bigl(-\tfrac12\,|x|_{c}^{2}-Z_{c}\Bigr).$$