Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $x_{k}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and integrability as in Measure Spaces and the Lebesgue Integral: Standing Notation, let $c$ be a variance sequence and $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$, and let $\exp$ be the exponential function.

1. (Coordinates)¶ For all $j,k\in\mathbb{N}$ the functions $x\mapsto x_{k}$ and $x\mapsto x_{j}x_{k}$ are Borel and integrable with respect to $\gamma_{c}$, $\int_{X}x_{k}\,\gamma_{c}(dx)=0$, and

$$\int_{X}x_{j}x_{k}\,\gamma_{c}(dx)=\begin{cases}c_{k}&\text{if }j=k,\\0&\text{if }j\ne k.\end{cases}$$

2. (Second moment)¶ $\gamma_{c}\in\mathcal{P}_{2}(X)$ in the sense of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, and its second moment is $M_{2}(\gamma_{c})=\sum_{k=1}^{\infty}c_{k}$.

3. (Exponential moments)¶ Let $\alpha,\theta\in\mathbb{R}$ with $0\le\alpha$ and $\theta<1$, and suppose that $2\alpha c_{k}\le\theta$ for every $k\in\mathbb{N}$. Then the function $x\mapsto\exp(\alpha|x|^{2})$ is Borel and integrable with respect to $\gamma_{c}$, and

$$\int_{X}\exp\bigl(\alpha|x|^{2}\bigr)\,\gamma_{c}(dx)\le\exp\Bigl(\frac{\alpha}{1-\theta}\sum_{k=1}^{\infty}c_{k}\Bigr).$$