Proof of The Standard Normal Distribution Has Finite Fisher Information, Score Minus the Identity, and Fisher Information One

Each result cited is universally quantified over the data in its own statement. The identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative are in force: $\mathbb{R}=\mathbb{R}^{1}$, a point of $\mathbb{R}^{1}$ being its sole coordinate, $\mathcal{B}(\mathbb{R}^{1})=\mathcal{B}(\mathbb{R})$, so that $\lambda=\lambda_{1}$ is a measure on $\mathcal{B}(\mathbb{R})$ and $N\in\mathcal{P}(\mathbb{R})$, and $a\cdot z=az$ and $\lVert z\rVert^{2}=z^{2}$ for $a,z\in\mathbb{R}$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars). Field arithmetic in $\mathbb{R}$ is that of Field, a quotient $s/t$ for $t\ne0$ meaning $s\,t^{-1}$, and $2=1+1$. $d_{\mathbb{R}}$ is the absolute-value metric on $\mathbb{R}$, and the Euclidean distance $d_{E}$ of $\mathbb{R}^{1}$ is $d_{\mathbb{R}}$ by The Euclidean Distance on the Real Line is the Absolute Value Metric. Convergence of a sequence of real numbers is that of Limit of a Sequence of Real Numbers, read as stipulated in the preamble of Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, and convergence in a metric space is that of Convergent Sequence in a Metric Space. A nonnegative Borel function $\mathbb{R}\to\mathbb{R}$ is read as a $[0,\infty]$-valued map where such a map is required, the two readings of measurability agreeing by Lebesgue Integral of a Nonnegative Measurable Function. $\mathbb{R}^{1}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous.

Throughout, The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder is applied in dimension $1$ with smoothing parameter $1$ (in the role of its $m$ and its $\eta$, and $0<1\le1$), and $\psi_{1}$, $c_{1}$ and $\varphi_{1}=c_{1}\psi_{1}$ denote the objects of that lemma. For $z\in\mathbb{R}^{1}$, $\psi_{1}(z)=\exp\bigl(-\lVert z\rVert^{2}/(2\cdot1)\bigr)=\exp(-z^{2}/2)=g(z)$, so $\psi_{1}=g$; and $c_{1}$ is the unique real number $c_{1}>0$ with $\int_{\mathbb{R}}c_{1}g\,d\lambda=1$ (claim 1 there, $\lambda_{1}$ being $\lambda$ by Lebesgue Measure on $\mathbb{R}^n$).

Step 1: $\varphi=\varphi_{1}$. Since $c>0$, the inverse $c^{-1}$ exists and $c^{-1}>0$ by claim 7 of Elementary Order Arithmetic in an Ordered Field. By claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, $g=\psi_{1}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{1})$ and $\mathcal{B}(\mathbb{R})$ and $0<g(z)$ for every $z$, so by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, $\int_{\mathbb{R}}c^{-1}g\,d\lambda=c^{-1}\int_{\mathbb{R}}g\,d\lambda=c^{-1}c=1$. By the uniqueness of $c_{1}$, $c_{1}=c^{-1}$, and therefore $\varphi_{1}=c^{-1}g=\varphi$.

Step 2: claim 1. By claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, $\varphi=\varphi_{1}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{1})$ and $\mathcal{B}(\mathbb{R})$, that is, Borel in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, it is sequentially continuous, and $0<\varphi(z)$ for every $z$. By claim 2 there, applied with $i=1$, the partial derivative $\partial_{1}\varphi(z)$ exists at every $z\in\mathbb{R}^{1}$, with $\partial_{1}\varphi(z)=-\frac{z_{1}}{1}\varphi(z)=-z\,\varphi(z)$, and $\partial_{1}\varphi$ is sequentially continuous.

Sequential continuity implies continuity in the Euclidean sense. Let $h:\mathbb{R}^{1}\to\mathbb{R}$ be sequentially continuous in the sense of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder: $h(z^{k})\to h(z)$ as a sequence of real numbers whenever $(z^{k})_{k\in\mathbb{N}}$ is a sequence in $\mathbb{R}^{1}$ and $z\in\mathbb{R}^{1}$ with $d_{E}(z^{k},z)\to0$ as a sequence of real numbers. Let $z\in\mathbb{R}^{1}$ and let $(z^{k})_{k\in\mathbb{N}}$ converge to $z$ in the metric space $(\mathbb{R}^{1},d_{E})$: for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $d_{E}(z^{k},z)<\varepsilon$ for every $k\in\mathbb{N}$ with $k\ge N$. Since $0\le d_{E}(z^{k},z)$ (Euclidean Distance is a Metric on $\mathbb{R}^n$) we have $|d_{E}(z^{k},z)-0|=d_{E}(z^{k},z)$ by Absolute Value in an Ordered Field, so the same condition says that the real sequence $(d_{E}(z^{k},z))_{k}$ converges to $0$ in the metric space $(\mathbb{R},d_{\mathbb{R}})$, hence converges to $0$ as a sequence of real numbers by claim 1 of Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space. By sequential continuity, $h(z^{k})\to h(z)$ as a sequence of real numbers, hence in $(\mathbb{R},d_{\mathbb{R}})$ by claim 1 of Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space again. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion, applied with $X=A=\mathbb{R}^{1}$ (with $d_{E}$) and $Y=(\mathbb{R},d_{\mathbb{R}})$, $h$ is continuous at $z$ relative to $\mathbb{R}^{1}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$; and by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, applied with $n=1$ and $E=\mathbb{R}^{1}$, $h$ is continuous at $z$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces.

Applying this to $h=\varphi$ and to $h=\partial_{1}\varphi$, both are continuous at every point of $\mathbb{R}^{1}$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces; since $\mathbb{R}^{1}$ is open (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3, shows that $\varphi$ is of class $C^{1}$ on $\mathbb{R}^{1}$. By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $\varphi$ is continuous in the sense of the preamble of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, and it is Borel as recorded above. By claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $\varphi$ is differentiable at every $x\in\mathbb{R}$ with $\varphi'(x)=\partial_{1}\varphi(x)=-x\,\varphi(x)$.

The density. Let $B\in\mathcal{B}(\mathbb{R})$. By Standard Normal Distribution and claim 2 of The Gaussian Weight Defines a Probability Distribution, $N(B)=c^{-1}\int_{\mathbb{R}}\mathbf{1}_{B}\,g\,d\lambda$, the integral of the nonnegative function $\mathbf{1}_{B}\,g$, which is measurable by The Gaussian Weight Defines a Probability Distribution. By claim 1 of Linearity and Monotonicity of the Lebesgue Integral, with the constant $c^{-1}\in[0,\infty)$, $c^{-1}\int_{\mathbb{R}}\mathbf{1}_{B}\,g\,d\lambda=\int_{\mathbb{R}}c^{-1}\,\mathbf{1}_{B}\,g\,d\lambda$, and $c^{-1}\bigl(\mathbf{1}_{B}(x)\,g(x)\bigr)=\mathbf{1}_{B}(x)\bigl(c^{-1}g(x)\bigr)=\mathbf{1}_{B}(x)\,\varphi(x)$ for every $x$ by associativity and commutativity of multiplication. Hence $N(B)=\int_{\mathbb{R}}\mathbf{1}_{B}\,\varphi\,d\lambda$, which is $\nu_{\varphi}(B)$ for the measure $\nu_{\varphi}$ with density $\varphi$ with respect to $\lambda$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables, applied on $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ with $h=\varphi$, a Borel function with values in $[0,\infty)$. So $N=\nu_{\varphi}$.

Step 3: claim 2. By The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment, $M_{2}(N)=\int_{\mathbb{R}}\lVert x\rVert^{2}\,N(dx)=\int_{\mathbb{R}}x^{2}\,N(dx)$, the integral of the nonnegative Borel function $x\mapsto\lVert x\rVert^{2}=x^{2}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs), and by claim 3 of Image Measures, Measures with Densities, and Change of Variables and Step 2 this equals $\int_{\mathbb{R}}x^{2}\varphi(x)\,\lambda(dx)$ in $[0,\infty]$. Apply claim 4 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $a=1\in\mathbb{R}^{1}$ (and smoothing parameter $1$): there $Z_{a}(z)=(a\cdot z)/1=(1\cdot z)/1=z$ and $\kappa_{a}=\lVert a\rVert^{2}/1=1^{2}/1=1$, so the function $Z_{a}^{2}\varphi_{1}:z\mapsto z^{2}\varphi(z)$ is integrable with respect to $\lambda$, with Lebesgue integral $\int_{\mathbb{R}}z^{2}\varphi(z)\,\lambda(dz)=\kappa_{a}=1$ in the sense of Integrable Function and the Lebesgue Integral. This function is nonnegative: $0<\varphi(z)$ (Step 2), hence $0\le\varphi(z)$, and $0\le z^{2}$ by claim 2 of Nonnegativity of Squares in an Ordered Field, so $\varphi(z)\cdot0\le\varphi(z)z^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, and $\varphi(z)\cdot0=0$ by claim 1 of Zero Products and Elementary Identities in a Field. Hence its positive part is the function itself and its negative part is the zero function, which is the nonnegative simple function $\mathbf{1}_{\varnothing}$, whose integral is $\lambda(\varnothing)=0$ by The Integral of an Indicator Function is the Measure of the Set and Measure, Measure Space, and Probability Measure; so by Integrable Function and the Lebesgue Integral its Lebesgue integral equals its integral as a nonnegative function, and $M_{2}(N)=1$. In particular $M_{2}(N)<\infty$, so $N\in\mathcal{P}_{2}(\mathbb{R})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space.

Step 4: claim 3. Apply The Score of a Measure with a Positive Continuously Differentiable Density with $d=1$, with the density $\varphi$ in the role of its $m$, and with $\mu=N$: by Steps 2 and 3, $\varphi$ is of class $C^{1}$ on $\mathbb{R}^{1}$ with $\varphi(x)>0$ for every $x$, and $N\in\mathcal{P}_{2}(\mathbb{R})$ is the measure with density $\varphi$ with respect to $\lambda_{1}=\lambda$. Its gradient map is $\nabla\varphi(x)=D\varphi(x)$, the point of $\mathbb{R}^{1}$ with single coordinate $\partial_{1}\varphi(x)=-x\,\varphi(x)$ (Gradient of a Real-Valued Function on a Euclidean Open Set), so the vector field of that lemma, written $\eta$ there and $\zeta$ here, is $\zeta(x)=\varphi(x)^{-1}\nabla\varphi(x)$, the scalar multiple in $\mathbb{R}^{1}$ being formed on the single coordinate (Scalar Multiple of a Point of $\mathbb{R}^n$): $\zeta(x)=\varphi(x)^{-1}\bigl(-(x\,\varphi(x))\bigr)=-\bigl(\varphi(x)^{-1}\varphi(x)\,x\bigr)=-x$ by claim 2 of Zero Products and Elementary Identities in a Field and the field axioms. Thus $\zeta(x)=-x=(-1)x$ for every $x$, since $(-1)x=-(1x)=-x$ by claim 2 of Zero Products and Elementary Identities in a Field. The remaining hypothesis of the lemma holds: $\lVert\zeta(x)\rVert^{2}=(-x)^{2}=x^{2}$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars and claim 2 of Zero Products and Elementary Identities in a Field, so $\int_{\mathbb{R}}\lVert\zeta\rVert^{2}\,dN=M_{2}(N)=1<\infty$ by Step 3. By The Score of a Measure with a Positive Continuously Differentiable Density §projection, $N$ has finite Fisher information, so $N\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R})$. The class of $\zeta$ belongs to $L^{2}(N;\mathbb{R})$ by The Score of a Measure with a Positive Continuously Differentiable Density §identity; its representative is $x\mapsto(-1)x$, so it is the scalar multiple $(-1)\mathrm{id}$ of the class of $\mathrm{id}$, scalar multiples of classes being formed on representatives (The Space of Square-Integrable Random Vectors §classes), and $(-1)\mathrm{id}=-\mathrm{id}$ by claim 5 of Elementary Identities in a Vector Space. Since $\mathrm{id}\in T_{N}$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity and $T_{N}$ is a closed linear subspace of $L^{2}(N;\mathbb{R})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed, hence a linear subspace by Real Hilbert Space §closed-subspace, clause 3 of Linear Subspace gives $\zeta=-\mathrm{id}\in T_{N}$. By The Score of a Measure with a Positive Continuously Differentiable Density §equality, $\xi_{N}=\zeta=-\mathrm{id}$ and $\mathcal{I}(N)=\lVert\zeta\rVert_{N}^{2}=\int_{\mathbb{R}}\lVert\zeta\rVert^{2}\,dN=1$, the last two equalities by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars and Step 3.