Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, let $c$ be a variance sequence and $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$. The relative score with respect to $\gamma_{c}$ of a measure $\mu\in\mathcal{P}_{2}(X)$ and its components are those of that definition, with $L^{2}(\mu)$ and its norm $\lVert\cdot\rVert_{L^{2}(\mu)}$ as there.

1. (Weight sequences)¶ A weight sequence is a sequence $a=(a_{k})_{k\in\mathbb{N}}$ of positive real numbers for which there is $\bar{a}\in\mathbb{R}$ with $a_{k}\le\bar{a}$ for every $k\in\mathbb{N}$.

2. (Weighted relative Fisher information)¶ Let $a$ be a weight sequence and let $\mu\in\mathcal{P}_{2}(X)$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$. The measure $\mu$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$ if the series $\sum_{k=1}^{\infty}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ converges, and its Fisher information relative to $\gamma_{c}$ with weights $a$ is then the sum

$$\mathcal{I}_{a}(\mu\,|\,\gamma_{c})=\sum_{k=1}^{\infty}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2},$$

a nonnegative real number by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, the terms being nonnegative.