Proof of Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation

Throughout, $\lambda_{k}$ is Lebesgue measure on $\mathbb{R}^{k}$ under the axiom of dependent choice, $c_{\mathrm{G}}$ is as in Standard Normal Distribution, and $\varphi$, $\gamma$, $\varphi_{k}$, $\gamma_{k}$ are as in Standard Normal Distribution (i), Standard Normal Distribution (ii), Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (i) and Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (ii); a measure $h\mu$ with a density gives a set $A$ the mass $\int\mathbf{1}_{A}h\,d\mu$, and integrals against it are integrals of the product with $h$ (Measure with a Density (i), Integrals over Measurable Subsets (i), Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vi), Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vii)); continuous and Borel functions of random vectors, their coordinates and their concatenations are random vectors, with laws the image laws (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (i), Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii), Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii), The Law of a Random Element (i)); and a finite product of elements of $[0,\infty)$ is the same in $\mathbb{R}$ and in $[0,\infty]$ (Addition and Multiplication on [0,∞] (ii), Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (ii)); every probability is finite, $P(A)\le P(\Omega)=1$ (Basic Properties of a Measure (iii)).

For Borel sets $B_{1},\dots,B_{k}\subseteq\mathbb{R}$ we write

$$B_{1}\times\dots\times B_{k}=\{x\in\mathbb{R}^{k}:x_{j}\in B_{j}\text{ for every }j\in[k]\},$$

and call these sets cylinders. A cylinder is the intersection of the sets $\pi_{j}^{-1}(B_{j})$, which are Borel by The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (iii) and Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), so, adding one coordinate at a time (Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1 (i)), it is Borel (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix)); its indicator is $x\mapsto\prod_{j=1}^{k}\mathbf{1}_{B_{j}}(x_{j})$.

Step 1 (the density is a product). Here and below, sums and products from $j=1$ to $k$ are taken over the finite set $[k]$ (Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v)). For $x\in\mathbb{R}^{k}$, $\lVert x\rVert^{2}=x\cdot x=\sum_{j=1}^{k}x_{j}^{2}$ (The Euclidean Norm on $\mathbb{R}^n$ (i), The Dot Product on $\mathbb{R}^n$ (i)). The map $u\mapsto\exp(-u/2)$ carries sums to products by Basic Properties of the Exponential Function (ii), so Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (ii) gives $\exp(-\lVert x\rVert^{2}/2)=\prod_{j=1}^{k}\exp(-x_{j}^{2}/2)$. Writing $c_{\mathrm{G}}^{-k}=(c_{\mathrm{G}}^{-1})^{k}$ (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (iv)) as the product of $k$ factors $c_{\mathrm{G}}^{-1}$ and combining the two products termwise (Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (v)),

$$\varphi_{k}(x)=\prod_{j=1}^{k}\varphi(x_{j})\qquad(x\in\mathbb{R}^{k}).$$

Step 2 ((i), and the measure of a cylinder). By The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments (i), $x\mapsto\exp(-x^{2}/2)$ is integrable with integral $c_{\mathrm{G}}$, so $\varphi$ is integrable and, taking the constant $c_{\mathrm{G}}^{-1}$ out of the integral (Basic Properties of the Integral of Nonnegative Measurable Maps (iv), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)),

$$\gamma(\mathbb{R})=\int_{\mathbb{R}}\varphi\,d\lambda=c_{\mathrm{G}}^{-1}\int_{\mathbb{R}}\exp(-x^{2}/2)\,d\lambda(x)=c_{\mathrm{G}}^{-1}c_{\mathrm{G}}=1 .$$

By Step 1 and The Lebesgue Integral over ℝⁿ of a Product of Functions of One Coordinate Each (i), $\gamma_{k}(\mathbb{R}^{k})$ is then a product of ones (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii)):

$$\gamma_{k}(\mathbb{R}^{k})=\int_{\mathbb{R}^{k}}\prod_{j=1}^{k}\varphi(x_{j})\,d\lambda_{k}(x)=\prod_{j=1}^{k}\int_{\mathbb{R}}\varphi\,d\lambda=1 .$$

Let $B_{1},\dots,B_{k}\in\mathcal{B}(\mathbb{R})$. By Step 1, the integrand of $\gamma_{k}(B_{1}\times\dots\times B_{k})=\int\mathbf{1}_{B_{1}\times\dots\times B_{k}}\varphi_{k}\,d\lambda_{k}$ is $x\mapsto\prod_{j=1}^{k}(\varphi\mathbf{1}_{B_{j}})(x_{j})$: on the cylinder both equal $\varphi_{k}(x)$, and off it both vanish, since some factor $\mathbf{1}_{B_{j}}(x_{j})$ is $0$ (Basic Properties of the Integral of Nonnegative Measurable Maps (ii)). So The Lebesgue Integral over ℝⁿ of a Product of Functions of One Coordinate Each (ii) and Standard Normal Distribution (ii) give

$$\gamma_{k}(B_{1}\times\dots\times B_{k})=\prod_{j=1}^{k}\int_{B_{j}}\varphi\,d\lambda=\prod_{j=1}^{k}\gamma(B_{j}).\tag{$\ast$}$$

As $\gamma$ and $\gamma_{k}$ are measures (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (v)), they are probability measures. Steps 1 and 2, and Step 3 below, hold in every dimension $k\in\mathbb{N}$.

Step 3 (cylinders determine a law). Let $\mu$ be a probability measure on $(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$ that agrees with $\gamma_{k}$ on every cylinder. Then $\mu=\gamma_{k}$. Indeed, let $\mathcal{C}_{k}$ be the set of cylinders. It is a $\pi$-system, since

$$(B_{1}\times\dots\times B_{k})\cap(B_{1}'\times\dots\times B_{k}')=(B_{1}\cap B_{1}')\times\dots\times(B_{k}\cap B_{k}').$$

Each set $\pi_{i}^{-1}(B)$ with $i\in[k]$ and $B\in\mathcal{B}(\mathbb{R})$ is the cylinder with $B$ in place $i$ and $\mathbb{R}$ elsewhere, so $\mathcal{B}(\mathbb{R}^{k})\subseteq\sigma(\mathcal{C}_{k})$ by The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (iii) and Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (v); conversely $\mathcal{C}_{k}\subseteq\mathcal{B}(\mathbb{R}^{k})$, so $\sigma(\mathcal{C}_{k})\subseteq\mathcal{B}(\mathbb{R}^{k})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv). As $\mu(\mathbb{R}^{k})=1=\gamma_{k}(\mathbb{R}^{k})$ (Step 2), Uniqueness of Measures Agreeing on a Generating π-System (i) gives $\mu=\gamma_{k}$.

Step 4 ((iii), "only if"). Let $Z$ be a standard normal vector: $Z_{1},\dots,Z_{k}$ are independent and $P_{Z_{j}}=\gamma$ for every $j$ (Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (iii), Standard Normal Distribution (iii)). For a cylinder $B_{1}\times\dots\times B_{k}$, Independent Random Elements: Preimages, Functions and Blocks (i), applied to the indices $1,\dots,k$, and $(\ast)$ give

$$P_{Z}(B_{1}\times\dots\times B_{k})=P(Z_{1}\in B_{1},\dots,Z_{k}\in B_{k})=\prod_{j=1}^{k}P(Z_{j}\in B_{j})=\prod_{j=1}^{k}\gamma(B_{j})=\gamma_{k}(B_{1}\times\dots\times B_{k}).$$

The law $P_{Z}$ is a probability measure, so $P_{Z}=\gamma_{k}$ by Step 3.

Step 5 ((iii), "if"). Let $P_{Z}=\gamma_{k}$. Let $n\in\mathbb{N}$, let $j_{1},\dots,j_{n}\in[k]$ be distinct and let $B_{1},\dots,B_{n}\in\mathcal{B}(\mathbb{R})$. Put $D_{j}=B_{i}$ if $j=j_{i}$, and $D_{j}=\mathbb{R}$ if $j$ is none of $j_{1},\dots,j_{n}$. Then $\{Z_{j_{1}}\in B_{1},\dots,Z_{j_{n}}\in B_{n}\}=\{Z\in D_{1}\times\dots\times D_{k}\}$, and $(\ast)$ gives

$$P(Z_{j_{1}}\in B_{1},\dots,Z_{j_{n}}\in B_{n})=\gamma_{k}(D_{1}\times\dots\times D_{k})=\prod_{j=1}^{k}\gamma(D_{j})=\prod_{i=1}^{n}\gamma(B_{i}),$$

since, with $J=\{j_{1},\dots,j_{n}\}$, the factors with $j\in[k]\setminus J$ equal $\gamma(\mathbb{R})=1$, and reindexing the product over $J$ by the bijection $i\mapsto j_{i}$ from $[n]$ onto $J$, with $\gamma(D_{j_{i}})=\gamma(B_{i})$, gives $\prod_{i=1}^{n}\gamma(B_{i})$ (Sums and Products over a Finite Set and over an Interval (iv), Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (ii), Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (iv), Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii), Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (iii)). With $n=1$ this says $P_{Z_{j}}=\gamma$, so every $Z_{j}$ is standard normal (Standard Normal Distribution (iii)). Hence the right-hand side is $\prod_{i=1}^{n}P(Z_{j_{i}}\in B_{i})$, and $Z_{1},\dots,Z_{k}$ are independent by Independent Random Elements: Preimages, Functions and Blocks (i). So $Z$ is a standard normal vector (Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (iii)).

Step 6 ((ii)). The identity function of $\mathbb{R}^{k}$ on the probability space $(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}),\gamma_{k})$ (Step 2) has law $\gamma_{k}$, so it is a standard normal vector by Step 5.

Step 7 (the law of a concatenation). Let $U$ and $V$ be independent random vectors in $\mathbb{R}^{k}$ and $\mathbb{R}^{l}$ with $P_{U}=\gamma_{k}$ and $P_{V}=\gamma_{l}$. Here, and in (vii), $(U,V)$ in $\mathbb{R}^{k+l}$ means the random vector $\iota\circ(U,V)$, $\omega\mapsto\iota(U(\omega),V(\omega))$, where $\iota:\mathbb{R}^{k}\times\mathbb{R}^{l}\to\mathbb{R}^{k+l}$ is the concatenation map of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (ii) (continuous, since $d_{E}(\iota(z),\iota(z'))\le 2\,d_{\times}(z,z')$ by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (vi)). Then $P_{(U,V)}=\gamma_{k+l}$. Indeed, let $D_{1},\dots,D_{k+l}\in\mathcal{B}(\mathbb{R})$ and $D=D_{1}\times\dots\times D_{k+l}$. By Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (i) and Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (ii), the coordinates of $\iota(\xi,\eta)$ are $\xi_{j}$ in place $j\in[k]$ and $\eta_{j}$ in place $k+j$ for $j\in[l]$, and every place in $[k+l]$ is of exactly one of these forms; hence

$$\{(U,V)\in D\}=\{U\in D_{1}\times\dots\times D_{k}\}\cap\{V\in D_{k+1}\times\dots\times D_{k+l}\},$$

where $D_{k+1}\times\dots\times D_{k+l}=\{\eta\in\mathbb{R}^{l}:\eta_{j}\in D_{k+j}\text{ for every }j\in[l]\}$. By the independence of $U$ and $V$ (Independent Random Elements: Preimages, Functions and Blocks (ii)), $(\ast)$ for $k$ and for $l$, and Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (iii),

$$P((U,V)\in D)=\gamma_{k}(D_{1}\times\dots\times D_{k})\,\gamma_{l}(D_{k+1}\times\dots\times D_{k+l})=\prod_{j=1}^{k}\gamma(D_{j})\prod_{j=1}^{l}\gamma(D_{k+j})=\prod_{j=1}^{k+l}\gamma(D_{j})=\gamma_{k+l}(D),$$

the last equality by $(\ast)$ for $k+l$. The law $P_{(U,V)}$ is a probability measure, so $P_{(U,V)}=\gamma_{k+l}$ by Step 3 for $k+l$.

Step 8 ((vii)). By (iii), $P_{Z}=\gamma_{k}$ and $P_{W}=\gamma_{l}$; by Step 7, $P_{(Z,W)}=\gamma_{k+l}$; by (iii) again, $(Z,W)$ is a standard normal vector in $\mathbb{R}^{k+l}$.

Step 9 ((iv)). Let $Z$ be a standard normal vector and $j\in[k]$, so that $P_{Z_{j}}=\gamma$. By The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments (ii) and Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii) (multiplying by $c_{\mathrm{G}}^{-1}$), $x\mapsto x\varphi(x)$ and $x\mapsto x^{2}\varphi(x)$ are integrable with respect to $\lambda$, with integrals $0$ and $1$. The map $x\mapsto x$ is the identity of $\mathbb{R}$, hence Borel (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)), and $x\mapsto x^{2}$ is continuous, hence Borel (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)); and $\varphi$ is real-valued, so Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vii) shows that they are integrable with respect to $\gamma$ and

$$\int_{\mathbb{R}}x\,d\gamma(x)=0,\qquad\int_{\mathbb{R}}x^{2}\,d\gamma(x)=1 .$$

By Expectations Computed from the Law (ii), $Z_{j}$ and $Z_{j}^{2}$ are integrable with

$$\mathbb{E}Z_{j}=0,\qquad\mathbb{E}Z_{j}^{2}=1 .$$

Hence $Z_{j}\in\mathcal{L}^{2}(P)$ (Elementary Properties of the p-Seminorm (ii), as the power $|Z_{j}|^{2}$ of Power-Integrable Functions and the p-Seminorm (i) is the two-factor product $Z_{j}Z_{j}=Z_{j}^{2}$, Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i)), so $Z\in\mathcal{L}^{2}(P;\mathbb{R}^{k})$ (Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (i)); $Z$ is integrable (Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)) with $\mathbb{E}Z=(\mathbb{E}Z_{1},\dots,\mathbb{E}Z_{k})=0$ (Expectation (iii)); and $\operatorname{Var}Z_{j}=\mathbb{E}Z_{j}^{2}-(\mathbb{E}Z_{j})^{2}=1$ (Rules for Expectations, Variances and Covariances (vii)). Let $i,j\in[k]$ with $i\neq j$. For Borel sets $B,B'\subseteq\mathbb{R}$, Independent Random Elements: Preimages, Functions and Blocks (i), applied to the two indices $i,j$, gives $P(Z_{i}\in B,\ Z_{j}\in B')=P(Z_{i}\in B)\,P(Z_{j}\in B')$, so $Z_{i}$ and $Z_{j}$ are independent by Independent Random Elements: Preimages, Functions and Blocks (ii), and $\operatorname{Cov}(Z_{i},Z_{j})=0$ by Expectations and Variances of Independent Random Variables (ii). Therefore $\operatorname{Cov}(Z)_{ij}=\operatorname{Cov}(Z_{i},Z_{j})=\delta_{ij}$ for all $i,j\in[k]$ (Variance and Covariance (iv), Variance and Covariance (ii)), that is, $\operatorname{Cov}(Z)=I_{k}$.

Step 10 ((v)). Let $Z$ be a standard normal vector, $Q\in\mathrm{O}(k)$ and $B\in\mathcal{B}(\mathbb{R}^{k})$. The matrix $Q$ is invertible with $|\det Q|=1$ (Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (i), Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (vi)), and $\lVert Qx\rVert=\lVert x\rVert$, so $\varphi_{k}(Qx)=\varphi_{k}(x)$ for every $x\in\mathbb{R}^{k}$ (Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (iii)). The map $T(x)=Qx$ is continuous and Borel (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), Linear Change of Variables for Lebesgue Measure on ℝⁿ (ii)), so $QZ=T\circ Z$ has the image law $P_{QZ}=P_{Z}\circ T^{-1}$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)), and $P_{Z}=\gamma_{k}$ by (iii). The change of variables for nonnegative functions (Linear Change of Variables for Lebesgue Measure on ℝⁿ (iii)), applied to $\mathbf{1}_{B}\varphi_{k}\in\mathcal{M}^{+}$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vi)), together with $\mathbf{1}_{T^{-1}(B)}=\mathbf{1}_{B}\circ T$ and $\varphi_{k}\circ T=\varphi_{k}$, therefore gives

$$P(QZ\in B)=\gamma_{k}(T^{-1}(B))=\int\mathbf{1}_{B}(Qx)\,\varphi_{k}(Qx)\,d\lambda_{k}(x)=|\det Q|^{-1}\int\mathbf{1}_{B}\,\varphi_{k}\,d\lambda_{k}=\gamma_{k}(B).$$

So $P_{QZ}=\gamma_{k}$, and $QZ$ is a standard normal vector by (iii).

Step 11 ((vi)). The rows $c_{i}=C^{\top}e_{i}\in\mathbb{R}^{k}$ ($i\in[r]$) of $C$ have coordinates $(c_{i})_{j}=C_{ij}$ (Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (vi)), so $(CZ)_{i}=c_{i}\cdot Z$ (Matrix-Vector Product (i)), and they satisfy $c_{i}\cdot c_{j}=(CC^{\top})_{ij}=\delta_{ij}$ (Product of Real Matrices (i), Transpose of a Real Matrix (i)).

First we extend them to $k$ pairwise orthogonal vectors of norm $1$. By induction on $m\in[k]$ (Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1 (i); the induction formula is predicative by The Graph Reading of a Formula Tree Is Predicative and Agrees with Its Reading, So the Reading Is Predicative (iii)) there are $u_{1},\dots,u_{m}\in\mathbb{R}^{k}$ with $u_{i}\cdot u_{j}=\delta_{ij}$ for all $i,j\le m$ and $u_{i}=c_{i}$ for $i\le\min(r,m)$: start with $u_{1}=c_{1}$, and, given $u_{1},\dots,u_{m}$ with $m<k$, put $u_{m+1}=c_{m+1}$ if $m+1\le r$ (as $c_{m+1}\cdot c_{m+1}=1$ and $c_{m+1}\cdot u_{i}=c_{m+1}\cdot c_{i}=0$ for $i\le m$), and otherwise extend by Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (vii). This gives orthonormal $u_{1},\dots,u_{k}$ with $u_{i}=c_{i}$ for $i\le\min(r,k)$. The matrix $R$ with columns $u_{1},\dots,u_{k}$ lies in $\mathrm{O}(k)$ (Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (v)), hence so does $Q=R^{\top}$ (Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (i)); its entries are $Q_{ij}=(u_{i})_{j}$ (Transpose of a Real Matrix (i)), so $(Qx)_{i}=u_{i}\cdot x$ for $x\in\mathbb{R}^{k}$ and $i\in[k]$ (Matrix-Vector Product (i)).

Next, $r\le k$. Otherwise $Qc_{k+1}$ has the coordinates $u_{i}\cdot c_{k+1}=c_{i}\cdot c_{k+1}=0$ ($i\in[k]$), so $Qc_{k+1}=0$, and since $Q$ is an isometry (Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion (iii), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (vi))

$$c_{k+1}\cdot c_{k+1}=(Qc_{k+1})\cdot(Qc_{k+1})=0,$$

against $c_{k+1}\cdot c_{k+1}=1$. Hence $u_{i}=c_{i}$ for every $i\in[r]$, and

$$(QZ)_{i}=c_{i}\cdot Z=(CZ)_{i}\qquad(i\in[r]).$$

Finally, $P_{QZ}=\gamma_{k}$ by Step 10. Let $B_{1},\dots,B_{r}\in\mathcal{B}(\mathbb{R})$, and put $D_{i}=B_{i}$ for $i\in[r]$ and $D_{i}=\mathbb{R}$ for $r<i\le k$. Then $\{CZ\in B_{1}\times\dots\times B_{r}\}=\{QZ\in D_{1}\times\dots\times D_{k}\}$, so the computation of Step 5, with $QZ$ in place of $Z$, $n=r$ and $j_{i}=i$, and $(\ast)$ in dimension $r$ give

$$P(CZ\in B_{1}\times\dots\times B_{r})=\gamma_{k}(D_{1}\times\dots\times D_{k})=\prod_{i=1}^{r}\gamma(B_{i})=\gamma_{r}(B_{1}\times\dots\times B_{r}).$$

As $CZ$ is a random vector (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (i)), Step 3 in dimension $r$ gives $P_{CZ}=\gamma_{r}$, and $CZ$ is a standard normal vector in $\mathbb{R}^{r}$ by (iii) in dimension $r$.