Proof of The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information

Each result cited is universally quantified over the data in its own statement. The notation of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force. The noise weights $a$ form a weight sequence by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, so every $a_{k}$ is positive; $c$ is a variance sequence (A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian), and its truncation is $c^{(n)}=(c_{1},\dots,c_{n})$ by Variance Sequences and Their Truncations §truncations. We apply Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs to $c$, to $\mu$ and to the weight sequence $a$, and use its notation: for $n\in\mathbb{N}$, $\mu_{n}=(p_{n})_{\#}\mu$, which lies in $\mathcal{P}_{2}(\mathbb{R}^{n})$ by Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §pushforward; when $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$, $\zeta^{(n)}_{k}\in L^{2}(\mu_{n})$ ($k\in[n]$) are the components of its relative score; $\mathcal{I}^{(n)}_{a}(\mu_{n})$ is as there; and for $H\in L^{2}(\mu_{n})$, $H\circ p_{n}\in V_{n}$ is the class of Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §isometry. The space $V_{n}$ of the statement is the one of that lemma, both being formed from $\mu$, so $\Pi_{n}$ is the orthogonal projection onto $V_{n}$, as in Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §projections. Also $\tilde{\mu}_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head. When $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$ we write $\tilde{\zeta}^{(n)}=\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}$ and $\tilde{\zeta}^{(n)}_{k}\in L^{2}(\tilde{\mu}_{n})$ for its components, as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §euclidean; $\tilde{c}_{k}=c_{k}/a_{k}$ denotes the $k$th entry of $\tilde{c}^{(n)}$, so that $1/\tilde{c}_{k}=a_{k}/c_{k}$. In every space $L^{2}(\lambda)$ that occurs, the inner product of two classes is the integral of the product of representatives, which is integrable, and the norm is the square root of the integral of the square of a representative (The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product). Borel maps between Euclidean spaces are as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps: a real-valued map on $\mathbb{R}^{n}$ that is continuous for the Euclidean distance and the absolute-value metric is Borel, a map into $\mathbb{R}^{n}$ is Borel when its components are, and composites of Borel maps are Borel; sums, real multiples and products of Borel real-valued functions are Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By Continuity at a Point for Maps Between Euclidean Spaces, read with target $\mathbb{R}^{1}$, a function $f:\mathbb{R}^{n}\to\mathbb{R}$ is continuous at every point exactly when it is continuous from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with the absolute-value metric, the defining inequalities reading $d_{E}(x,y)<\delta$ and $|f(x)-f(y)|<\varepsilon$ (The Euclidean Distance on the Real Line is the Absolute Value Metric).

Step 1 (Diagonal rescalings). Let $b=(b_{1},\dots,b_{n})$ be an $n$-tuple of positive real numbers and let $D_{b}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the map $D_{b}(u)=(b_{1}u_{1},\dots,b_{n}u_{n})$. With $b^{-1}=(b_{1}^{-1},\dots,b_{n}^{-1})$ one has $D_{b^{-1}}\circ D_{b}=D_{b}\circ D_{b^{-1}}=\mathrm{id}$. Put $\beta=b_{1}+\dots+b_{n}$, a positive number; since $|b_{i}(u_{i}-v_{i})|\le b_{i}\lVert u-v\rVert$ for every $i$, one has $\lVert D_{b}u-D_{b}v\rVert\le\beta\lVert u-v\rVert$ for all $u,v\in\mathbb{R}^{n}$. Each component $u\mapsto b_{i}u_{i}$ is therefore continuous, and $D_{b}$ is Borel. We record four facts, for a function $f:\mathbb{R}^{n}\to\mathbb{R}$, a point $u\in\mathbb{R}^{n}$ and $i\in[n]$.

(a) If $f$ is continuous at $D_{b}u$, then $f\circ D_{b}$ is continuous at $u$: given $\varepsilon>0$, take $\eta>0$ with $|f(y)-f(D_{b}u)|<\varepsilon$ whenever $\lVert y-D_{b}u\rVert<\eta$; then $\lVert v-u\rVert<\eta/\beta$ gives $\lVert D_{b}v-D_{b}u\rVert<\eta$, hence $|f(D_{b}v)-f(D_{b}u)|<\varepsilon$.

(b) If the partial derivative of $f$ with respect to the $i$th variable exists at $D_{b}u$ (Partial Derivative on a Euclidean Open Set), then that of $f\circ D_{b}$ exists at $u$ and $\partial_{i}(f\circ D_{b})(u)=b_{i}\,\partial_{i}f(D_{b}u)$. Indeed, for real $s$, the point obtained from $u$ by adding $s$ to its $i$th entry is mapped by $D_{b}$ to the point obtained from $D_{b}u$ by adding $t=b_{i}s$ to its $i$th entry; so for $s\ne0$ the difference quotient of $f\circ D_{b}$ at $u$ with increment $s$ is $b_{i}$ times the difference quotient of $f$ at $D_{b}u$ with increment $t$. Given $\varepsilon>0$, take $\eta>0$ such that the latter quotient lies within $\varepsilon/b_{i}$ of $\partial_{i}f(D_{b}u)$ whenever $0<|t|<\eta$; then $0<|s|<\eta/b_{i}$ gives $0<|t|<\eta$, and the former quotient lies within $\varepsilon$ of $b_{i}\,\partial_{i}f(D_{b}u)$. The domain being $\mathbb{R}^{n}$, the membership condition of that definition holds trivially.

(c) If $f\in C^{1}_{b}(\mathbb{R}^{n})$, the set of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=n$, then $f\circ D_{b}\in C^{1}_{b}(\mathbb{R}^{n})$ and $\partial_{i}(f\circ D_{b})=b_{i}\,(\partial_{i}f)\circ D_{b}$ for every $i\in[n]$. By clause 1 of C^k Maps on a Euclidean Open Set, $f$ and every $\partial_{i}f$ are continuous at every point; by (a) and (b), $f\circ D_{b}$ is continuous at every point and its partial derivatives exist everywhere and equal $b_{i}(\partial_{i}f)\circ D_{b}$, which is continuous at every point by (a), a constant multiple of a function continuous at a point being continuous there. So $f\circ D_{b}$ is of class $C^{1}$ by that clause, and it and its partial derivatives are bounded, by a bound of $f$ and by $b_{i}$ times a bound of $\partial_{i}f$ respectively.

(d) If $f\in C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$, then $f\circ D_{b}\in C^{2}_{b}(\mathbb{R}^{n})$, with $\partial_{i}(f\circ D_{b})=b_{i}(\partial_{i}f)\circ D_{b}$ and $\partial_{j}\partial_{i}(f\circ D_{b})=b_{j}b_{i}(\partial_{j}\partial_{i}f)\circ D_{b}$ for $i,j\in[n]$. By clause 2 of C^k Maps on a Euclidean Open Set, $f$ is of class $C^{1}$ and every $\partial_{i}f$ is of class $C^{1}$, and $f$, $\partial_{i}f$ and $\partial_{j}\partial_{i}f$ are bounded; so $f\in C^{1}_{b}(\mathbb{R}^{n})$ and $b_{i}\partial_{i}f\in C^{1}_{b}(\mathbb{R}^{n})$, with $\partial_{j}(b_{i}\partial_{i}f)=b_{i}\partial_{j}\partial_{i}f$, the difference quotients of $b_{i}\partial_{i}f$ being $b_{i}$ times those of $\partial_{i}f$. By (c), $f\circ D_{b}$ is of class $C^{1}$ with $\partial_{i}(f\circ D_{b})=(b_{i}\partial_{i}f)\circ D_{b}$, and by (c) again this function is of class $C^{1}$ with $j$th partial derivative $b_{j}b_{i}(\partial_{j}\partial_{i}f)\circ D_{b}$; hence $f\circ D_{b}$ is of class $C^{2}$ by clause 2 of C^k Maps on a Euclidean Open Set, and the bounds follow as in (c).

The map $\delta_{n}$ of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads is $D_{b}$ with $b_{k}=a_{k}^{-1/2}$, and its inverse is $\delta_{n}^{-1}=D_{b^{-1}}$, $\delta_{n}^{-1}(y)=(a_{1}^{1/2}y_{1},\dots,a_{n}^{1/2}y_{n})$. Both are Borel bijections of $\mathbb{R}^{n}$, and by (c) the maps $\psi\mapsto\psi\circ\delta_{n}$ and $\psi\mapsto\psi\circ\delta_{n}^{-1}$ send $C^{1}_{b}(\mathbb{R}^{n})$ into itself and are mutually inverse.

Step 2 (The rescaled head as a push-forward of the projected measure). Let $n\in\mathbb{N}$. Since $r_{n}=\delta_{n}\circ p_{n}$ (A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads), for every $B\in\mathcal{B}(\mathbb{R}^{n})$ one has $\tilde{\mu}_{n}(B)=\mu(p_{n}^{-1}(\delta_{n}^{-1}(B)))=\mu_{n}(\delta_{n}^{-1}(B))$ by the definition of image measures (claim 1 of Image Measures, Measures with Densities, and Change of Variables, as fixed in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward); that is, $\tilde{\mu}_{n}=(\delta_{n})_{\#}\mu_{n}$ in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, whose change-of-variables formula is used below. Consequently:

(a) For Borel $G:\mathbb{R}^{n}\to\mathbb{R}$, $\int G^{2}\,d\tilde{\mu}_{n}=\int(G\circ\delta_{n})^{2}\,d\mu_{n}$ in $[0,\infty]$; so if $G$ is $2$-integrable with respect to $\tilde{\mu}_{n}$, then $G\circ\delta_{n}$ is $2$-integrable with respect to $\mu_{n}$ and the two classes have the same norm. If $G=G'$ outside a $\tilde{\mu}_{n}$-null Borel set $N$, then $G\circ\delta_{n}=G'\circ\delta_{n}$ outside $\delta_{n}^{-1}(N)$, and $\mu_{n}(\delta_{n}^{-1}(N))=\tilde{\mu}_{n}(N)=0$.

(b) For $G,G'$ that are $2$-integrable with respect to $\tilde{\mu}_{n}$, the product $GG'$ is integrable with respect to $\tilde{\mu}_{n}$ (The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product), so $\int GG'\,d\tilde{\mu}_{n}=\int(G\circ\delta_{n})(G'\circ\delta_{n})\,d\mu_{n}$.

(c) Let $\tilde{\psi}\in C^{1}_{b}(\mathbb{R}^{n})$ and $\psi=\tilde{\psi}\circ\delta_{n}$, which lies in $C^{1}_{b}(\mathbb{R}^{n})$ with $\partial_{k}\psi=a_{k}^{-1/2}(\partial_{k}\tilde{\psi})\circ\delta_{n}$ by Step 1(c), and let $k\in[n]$. Then

$$\int_{\mathbb{R}^{n}}\Bigl(\frac{y_{k}}{\tilde{c}_{k}}\tilde{\psi}(y)-\partial_{k}\tilde{\psi}(y)\Bigr)\tilde{\mu}_{n}(dy)=a_{k}^{1/2}\int_{\mathbb{R}^{n}}\Bigl(\frac{u_{k}}{c_{k}}\psi(u)-\partial_{k}\psi(u)\Bigr)\mu_{n}(du).\qquad(\ast)$$

Indeed, write $\tilde{F}$ and $F$ for the two integrands; they are integrable with respect to $\tilde{\mu}_{n}$ and to $\mu_{n}$ respectively by Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §integrable, applied with the measure $\tilde{\mu}_{n}$ and the variance vector $\tilde{c}^{(n)}$, and with the measure $\mu_{n}$ and the variance vector $c^{(n)}$. For $u\in\mathbb{R}^{n}$ one has $(\delta_{n}u)_{k}=a_{k}^{-1/2}u_{k}$, $\tilde{\psi}(\delta_{n}u)=\psi(u)$ and $\partial_{k}\tilde{\psi}(\delta_{n}u)=a_{k}^{1/2}\partial_{k}\psi(u)$, so $\tilde{F}(\delta_{n}u)=a_{k}^{1/2}F(u)$; the change of variables and the linearity of the integral (Linearity and Monotonicity of the Lebesgue Integral §integrable) give $(\ast)$.

Step 3 (Transfer of finite Fisher information). Let $n\in\mathbb{N}$.

(i) Suppose that $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$; let $k\in[n]$ and let $G_{k}$ be a Borel representative of $\zeta^{(n)}_{k}$. Put $\tilde{g}_{k}=a_{k}^{1/2}\,G_{k}\circ\delta_{n}^{-1}$, a Borel function with $\tilde{g}_{k}\circ\delta_{n}=a_{k}^{1/2}G_{k}$; by Step 2(a), $\int\tilde{g}_{k}^{2}\,d\tilde{\mu}_{n}=a_{k}\int G_{k}^{2}\,d\mu_{n}<\infty$, so $\tilde{g}_{k}$ has a class in $L^{2}(\tilde{\mu}_{n})$. Let $\tilde{\psi}\in C^{1}_{b}(\mathbb{R}^{n})$, which is $2$-integrable with respect to $\tilde{\mu}_{n}$ by Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §integrable, and let $\psi=\tilde{\psi}\circ\delta_{n}$. By Step 2(b), by the identity of claim 2 of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions, which $\zeta^{(n)}_{k}$ satisfies by Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §agreement applied with the measure $\mu_{n}$ and the variance vector $c^{(n)}$, and by $(\ast)$,

$$\langle\tilde{g}_{k},\tilde{\psi}\rangle_{L^{2}(\tilde{\mu}_{n})}=a_{k}^{1/2}\int G_{k}\psi\,d\mu_{n}=a_{k}^{1/2}\langle\zeta^{(n)}_{k},\psi\rangle_{L^{2}(\mu_{n})}=a_{k}^{1/2}\int\Bigl(\frac{u_{k}}{c_{k}}\psi(u)-\partial_{k}\psi(u)\Bigr)\mu_{n}(du)=\int\Bigl(\frac{y_{k}}{\tilde{c}_{k}}\tilde{\psi}(y)-\partial_{k}\tilde{\psi}(y)\Bigr)\tilde{\mu}_{n}(dy).$$

Do this for every $k\in[n]$: for each such $k$ the class of $\tilde{g}_{k}$ in $L^{2}(\tilde{\mu}_{n})$ satisfies the identity of claim 2 of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions for the index $k$ and every $\tilde{\psi}\in C^{1}_{b}(\mathbb{R}^{n})$. By Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §componentwise, applied with the measure $\tilde{\mu}_{n}$ and the variance vector $\tilde{c}^{(n)}$, the measure $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$, and by Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §agreement the class of $\tilde{g}_{k}$ is $\tilde{\zeta}^{(n)}_{k}$. Hence every Borel representative $g_{k}$ of $\tilde{\zeta}^{(n)}_{k}$ equals $\tilde{g}_{k}$ outside a $\tilde{\mu}_{n}$-null Borel set, and by Step 2(a), $g_{k}\circ\delta_{n}=a_{k}^{1/2}G_{k}$ outside a $\mu_{n}$-null Borel set.

(ii) Suppose that $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$; let $k\in[n]$ and let $g_{k}$ be a Borel representative of $\tilde{\zeta}^{(n)}_{k}$. Put $G_{k}=a_{k}^{-1/2}\,g_{k}\circ\delta_{n}$, a Borel function with $\int G_{k}^{2}\,d\mu_{n}=a_{k}^{-1}\int g_{k}^{2}\,d\tilde{\mu}_{n}<\infty$ by Step 2(a). Let $\psi\in C^{1}_{b}(\mathbb{R}^{n})$ and $\tilde{\psi}=\psi\circ\delta_{n}^{-1}$, which lies in $C^{1}_{b}(\mathbb{R}^{n})$ by Step 1, so that $\psi=\tilde{\psi}\circ\delta_{n}$. By Step 2(b), by the identity of claim 2 of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions, which $\tilde{\zeta}^{(n)}_{k}$ satisfies by Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §agreement applied with the measure $\tilde{\mu}_{n}$ and the variance vector $\tilde{c}^{(n)}$, and by $(\ast)$,

$$\langle G_{k},\psi\rangle_{L^{2}(\mu_{n})}=a_{k}^{-1/2}\int g_{k}\tilde{\psi}\,d\tilde{\mu}_{n}=a_{k}^{-1/2}\int\Bigl(\frac{y_{k}}{\tilde{c}_{k}}\tilde{\psi}(y)-\partial_{k}\tilde{\psi}(y)\Bigr)\tilde{\mu}_{n}(dy)=\int\Bigl(\frac{u_{k}}{c_{k}}\psi(u)-\partial_{k}\psi(u)\Bigr)\mu_{n}(du).$$

Do this for every $k\in[n]$, so that the identity of claim 2 of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions holds for every index. By Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §componentwise and Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §agreement, applied with the measure $\mu_{n}$ and the variance vector $c^{(n)}$, the measure $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$, and the class of $G_{k}$ is $\zeta^{(n)}_{k}$.

(iii) In either case both measures have finite Fisher information, and for $k\in[n]$, Borel representatives $g_{k}$ of $\tilde{\zeta}^{(n)}_{k}$ and $G_{k}$ of $\zeta^{(n)}_{k}$, the functions $g_{k}\circ\delta_{n}$ and $a_{k}^{1/2}G_{k}$ have the same class in $L^{2}(\mu_{n})$: by (i), or by (ii) together with the independence of classes from representatives in Step 2(a). Step 2(a) then gives $\lVert\tilde{\zeta}^{(n)}_{k}\rVert_{L^{2}(\tilde{\mu}_{n})}^{2}=\int(g_{k}\circ\delta_{n})^{2}\,d\mu_{n}=a_{k}\lVert\zeta^{(n)}_{k}\rVert_{L^{2}(\mu_{n})}^{2}$, so by Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §agreement

$$\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})=\sum_{k=1}^{n}\lVert\tilde{\zeta}^{(n)}_{k}\rVert_{L^{2}(\tilde{\mu}_{n})}^{2}=\sum_{k=1}^{n}a_{k}\lVert\zeta^{(n)}_{k}\rVert_{L^{2}(\mu_{n})}^{2}.$$

Moreover, if $\zeta=(\zeta_{1},\dots,\zeta_{n})$ is a representative of the relative score of $\mu_{n}$ with respect to $\gamma_{c^{(n)}}$, then $\zeta_{k}$ is a representative of $\zeta^{(n)}_{k}$ (preamble of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions), so by The Weighted Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space §information, with the weights $a^{(n)}=(a_{1},\dots,a_{n})$, and the additivity and homogeneity of the integral of nonnegative functions (Linearity and Monotonicity of the Lebesgue Integral §nonnegative),

$$\mathcal{I}^{(n)}_{a}(\mu_{n})=\int\sum_{k=1}^{n}a_{k}\zeta_{k}^{2}\,d\mu_{n}=\sum_{k=1}^{n}a_{k}\lVert\zeta^{(n)}_{k}\rVert_{L^{2}(\mu_{n})}^{2}=\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n}).$$

Step 4 (Claim 1). Let $\mu$ have the relative score $(\zeta_{k})_{k\in\mathbb{N}}$, and let $n\in\mathbb{N}$ and $k\in[n]$. By Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §projections, $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$ and $\Pi_{n}\zeta_{k}=\zeta^{(n)}_{k}\circ p_{n}$, which is the class in $L^{2}(\mu)$ of $G_{k}\circ p_{n}$ for a Borel representative $G_{k}$ of $\zeta^{(n)}_{k}$ (Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §isometry). By Step 3(i), $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$. Let $g_{k}$ be a Borel representative of $\tilde{\zeta}^{(n)}_{k}$. The function $a_{k}^{-1/2}(g_{k}\circ r_{n})$ is Borel, $r_{n}$ being Borel by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head, and $\int_{X}(a_{k}^{-1/2}g_{k}\circ r_{n})^{2}\,d\mu=a_{k}^{-1}\int g_{k}^{2}\,d\tilde{\mu}_{n}<\infty$ by the change-of-variables formula of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. By Step 3(i) there is a $\mu_{n}$-null Borel set $N$ outside which $g_{k}\circ\delta_{n}=a_{k}^{1/2}G_{k}$; since $r_{n}=\delta_{n}\circ p_{n}$, the functions $a_{k}^{-1/2}(g_{k}\circ r_{n})$ and $G_{k}\circ p_{n}$ agree outside $p_{n}^{-1}(N)$, and $\mu(p_{n}^{-1}(N))=\mu_{n}(N)=0$. So the class of $a_{k}^{-1/2}(g_{k}\circ r_{n})$ is the class of $G_{k}\circ p_{n}$, namely $\Pi_{n}\zeta_{k}$, whatever the choice of $g_{k}$. This proves claim 1.

Step 5 (Claim 2). In the situation of claim 1, Step 3(iii) gives $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})=\sum_{k=1}^{n}a_{k}\lVert\zeta^{(n)}_{k}\rVert_{L^{2}(\mu_{n})}^{2}$, and by Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §isometry and Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §projections, $\lVert\zeta^{(n)}_{k}\rVert_{L^{2}(\mu_{n})}=\lVert\zeta^{(n)}_{k}\circ p_{n}\rVert_{L^{2}(\mu)}=\lVert\Pi_{n}\zeta_{k}\rVert_{L^{2}(\mu)}$ for each $k\in[n]$. This proves claim 2.

Step 6 (Claim 3). Let $\mu$ have a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. For each $n\in\mathbb{N}$, $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$ by Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §projections, so Step 3(iii) gives $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})=\mathcal{I}^{(n)}_{a}(\mu_{n})$. Both assertions of claim 3 are therefore those of Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §convergence, applied with the weight sequence $a$.

Step 7 (Claim 4). Let $h\in C_{c}^{\infty}(\mathbb{R}^{n})$. It is smooth, hence of class $C^{1}$ and of class $C^{2}$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient). By The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient, for $i\in[n]$ the function $\partial_{i}h$ is continuous, compactly supported and bounded; it is smooth by claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map; so $\partial_{i}h\in C_{c}^{\infty}(\mathbb{R}^{n})$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space), and the same claim of The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure, applied to the test function $\partial_{i}h$, shows that $\partial_{j}\partial_{i}h$ is continuous and bounded for every $j\in[n]$. The function $h$ itself is continuous at every point (clause 1 of C^k Maps on a Euclidean Open Set), hence continuous from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$, and it is compactly supported for the topology of the open subsets of $\mathbb{R}^{n}$, which is the metric topology of $d_{E}$ by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$; so $h$ is bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable. Thus $h\in C^{2}_{b}(\mathbb{R}^{n})$, and Step 1(d), applied with $D_{b}=\delta_{n}$, gives $g=h\circ\delta_{n}\in C^{2}_{b}(\mathbb{R}^{n})$ with

$$\partial_{k}g=a_{k}^{-1/2}(\partial_{k}h)\circ\delta_{n},\qquad\partial_{k}\partial_{k}g=a_{k}^{-1}(\partial_{k}\partial_{k}h)\circ\delta_{n}\qquad(k\in[n]).$$

Moreover $h\circ r_{n}=h\circ\delta_{n}\circ p_{n}=g\circ p_{n}$.

The functional. For $k\in[n]$ let $F_{k}:\mathbb{R}^{n}\to\mathbb{R}$ be the function $F_{k}(y)=\frac{y_{k}}{\tilde{c}_{k}}\partial_{k}h(y)-\partial_{k}\partial_{k}h(y)$. It is Borel: the coordinate $y\mapsto y_{k}$ is Borel by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\partial_{k}h$ and $\partial_{k}\partial_{k}h$ are continuous. For $x\in X$ one has $(r_{n}x)_{k}=a_{k}^{-1/2}x_{k}$ and $1/\tilde{c}_{k}=a_{k}/c_{k}$, so

$$a_{k}\Bigl(\frac{x_{k}}{c_{k}}\partial_{k}g(p_{n}(x))-\partial_{k}\partial_{k}g(p_{n}(x))\Bigr)=\frac{a_{k}^{1/2}x_{k}}{c_{k}}\partial_{k}h(r_{n}x)-\partial_{k}\partial_{k}h(r_{n}x)=F_{k}(r_{n}x).$$

The function in parentheses is integrable with respect to $\mu$ by the preamble of The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates, so $F_{k}\circ r_{n}$ is integrable with respect to $\mu$; hence $F_{k}$ is integrable with respect to $\tilde{\mu}_{n}=(r_{n})_{\#}\mu$ and $\int F_{k}\,d\tilde{\mu}_{n}=\int_{X}F_{k}\circ r_{n}\,d\mu$, by the change-of-variables formula of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. Summing over $k\in[n]$, The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional gives $L^{a}_{\mu}(g)=\sum_{k=1}^{n}\int F_{k}\,d\tilde{\mu}_{n}$. Each $\partial_{k}\partial_{k}h$ is bounded and Borel, hence integrable with respect to $\tilde{\mu}_{n}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), so the function $y\mapsto\frac{y_{k}}{\tilde{c}_{k}}\partial_{k}h(y)=F_{k}(y)+\partial_{k}\partial_{k}h(y)$ is integrable as well. The components of the scaling map $S_{\tilde{c}^{(n)}}$ are the maps $y\mapsto y_{k}/\tilde{c}_{k}$ (preamble of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information) and those of $\nabla h$ are the $\partial_{k}h$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient), so $\sum_{k=1}^{n}\frac{y_{k}}{\tilde{c}_{k}}\partial_{k}h(y)=S_{\tilde{c}^{(n)}}(y)\cdot\nabla h(y)$, while $\sum_{k=1}^{n}\partial_{k}\partial_{k}h=\Delta h$ (The Laplacian of a Twice Continuously Differentiable Function §laplacian). By the linearity of the integral (Linearity and Monotonicity of the Lebesgue Integral §integrable) and the inner product of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu,

$$L^{a}_{\mu}(g)=\int S_{\tilde{c}^{(n)}}\cdot\nabla h\,d\tilde{\mu}_{n}-\int\Delta h\,d\tilde{\mu}_{n}=\langle S_{\tilde{c}^{(n)}},\nabla h\rangle_{\tilde{\mu}_{n}}-\int\Delta h\,d\tilde{\mu}_{n}=-\ell^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}(h),$$

the last equality being Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional.

The norm. By Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §gradient, applied with $\psi=h$ (the measure written $\tilde{\mu}$ there being $\tilde{\mu}_{n}$), $\nabla_{a}(g\circ p_{n})(x)=\nabla_{a}(h\circ r_{n})(x)=\Lambda_{n}(\nabla h)(x)$ for every $x\in X$, so the two maps have the same class in $L^{2}(\mu;X^{a})$. The gradient map $\nabla h$ is Borel with $\int\lVert\nabla h\rVert^{2}\,d\tilde{\mu}_{n}<\infty$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test), so Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift gives $\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}^{2}=\langle\Lambda_{n}\nabla h,\Lambda_{n}\nabla h\rangle_{\mu}=\langle\nabla h,\nabla h\rangle_{\tilde{\mu}_{n}}=\lVert\nabla h\rVert_{\tilde{\mu}_{n}}^{2}$; both norms being nonnegative, they are equal. This proves claim 4.

Step 8 (Claim 5). Under the hypotheses of claim 5, Step 3(ii) shows that $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$ for every $n\in\mathbb{N}$, and Step 3(iii) gives $\mathcal{I}^{(n)}_{a}(\mu_{n})=\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\le C$ for every $n\in\mathbb{N}$. Claim 5 is therefore Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §converse, applied with the weight sequence $a$.