Proof of Wick Powers of Paley-Wiener Functionals: Expansion in Cylindrical Hermite Polynomials, the Covariance Formula, Continuity in the Vector, and Membership in the Chaos

Each result cited is universally quantified over the data in its own statement.

The coordinates are $x_{k}=\langle x,e_{k}\rangle$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates); $c$ is a variance sequence, so $c_{k}>0$ (Variance Sequences and Their Truncations §variances). We write $L^{2}=L^{2}(\gamma_{c})$, a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{L^{2}(\gamma_{c})}$ and norm $\lVert\cdot\rVert_{2}$ (Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue, The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert); convergence in $L^{2}$ refers to the distance $d(F,G)=\lVert F-G\rVert_{2}$ (Real Hilbert Space §topology). For $f\in H_{c}$ and $q\in\mathbb{N}_{0}$, $|f|_{c}^{2q}$ denotes $(|f|_{c}^{2})^{q}$. Limits of real sequences are combined by Arithmetic of Limits of Real Sequences §sums, Arithmetic of Limits of Real Sequences §products and Arithmetic of Limits of Real Sequences §scalar, referred to below as the limit laws, and are unique by uniqueness of limits. The limit laws extend to finite sums and to powers: if $p\in\mathbb{N}$ and $(r_{k,N})_{N\in\mathbb{N}}$ converges to $r_{k}$ for each $k\in[p]$, the set of $l\in\mathbb{N}$ such that, if $l\le p$, then $\bigl(\sum_{k=1}^{l}r_{k,N}\bigr)_{N}$ converges to $\sum_{k=1}^{l}r_{k}$ contains $1$ and contains $l+1$ whenever it contains $l$, by claim 1 of Properties of Finite Sums and the law for sums, so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers; and if $(r_{N})$ converges to $r$, the set of $q\in\mathbb{N}$ such that $(r_{N}^{q})_{N}$ converges to $r^{q}$ contains $1$ and contains $q+1$ whenever it contains $q$, by claim 1 of Properties of Natural Number Powers in a Field and the law for products, so it is $\mathbb{N}$ by the same axiom (for the exponent $0$ all terms are $1$).

Step 0 (Vectors with finitely many nonzero coordinates). Let $f\in X$ and $N\in\mathbb{N}$ with $f_{k}=0$ for $k>N$. For $n>N$ the $n$-th partial sum of $\sum_{k}f_{k}^{2}/c_{k}$ equals $\sum_{k=1}^{N}f_{k}^{2}/c_{k}$, by Splitting a Finite Sum at an Index (with $N$ and $n-N$ in place of $m$ and $n$ there), the second part being a sum of zeros and hence $0$ by claim 7 of Properties of Finite Sums; the same splitting is used for the finite sums below. So the series converges with this sum (Series of Real Numbers §convergent): $f\in H_{c}$ and $|f|_{c}^{2}=\sum_{k=1}^{N}f_{k}^{2}/c_{k}$ (The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §space, The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §square). In the same way, for $g\in H_{c}$, $\langle f,g\rangle_{c}=\sum_{k=1}^{N}f_{k}g_{k}/c_{k}$ (The Cameron-Martin Inner Product of Two Cameron-Martin Vectors §inner-product); and for every $x\in X$ the Paley-Wiener partial sums satisfy $\ell_{f,n}(x)=\sum_{k=1}^{N}f_{k}x_{k}/c_{k}$ for $n\ge N$ (The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §partial-sums), so the sequence converges and $\ell_{f}(x)=\sum_{k=1}^{N}(f_{k}/c_{k})x_{k}$ by The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §functional. For $h\in X$ and $N\in\mathbb{N}$ let $h^{[N]}=\sum_{i=1}^{N}h_{i}e_{i}$. By Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and orthonormality of the basis (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space, Orthonormal Basis of a Real Hilbert Space §basis, Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §orthonormal), $h^{[N]}_{k}=\sum_{i=1}^{N}h_{i}\langle e_{i},e_{k}\rangle$ equals $h_{k}$ for $k\le N$ and $0$ for $k>N$, by claim 7 of Properties of Finite Sums (for $k\le N$ the only possibly nonzero summand is the one with $i=k$; for $k>N$ all summands vanish). Hence $h^{[N]}\in H_{c}$, and $\ell_{h^{[N]}}(x)=\sum_{k=1}^{N}h_{k}x_{k}/c_{k}=\ell_{h,N}(x)$ for every $x\in X$.

Step 1 (A multinomial addition formula). Let $N\in\mathbb{N}$, $v_{1},\dots,v_{N}\ge0$, $u_{1},\dots,u_{N}\in\mathbb{R}$ and $n\in\mathbb{N}_{0}$, and let $T_{n,N}$ be the set of $a\in\mathbb{N}_{0}^{N}$ with $a_{1}+\dots+a_{N}=n$. It contains the tuple $(n,0,\dots,0)$, and each $a\in T_{n,N}$ has $a_{k}\le n$ for all $k$ by claim 6 of Properties of Finite Sums; so $T_{n,N}$ is a nonempty subset of the set of $N$-tuples in $\{0,\dots,n\}$, which is finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets ($\{0,\dots,n\}$ being the image of $[n+1]$ under $k\mapsto k-1$, so finite by claim 4 of Basic Properties of Finite Sets), and $T_{n,N}$ is finite by claim 3 of Basic Properties of Finite Sets. Sums over such finite index sets are those of Sum over a Finite Index Set. We show

$$H^{v_{1}+\dots+v_{N}}_{n}(u_{1}+\dots+u_{N})=\sum_{a\in T_{n,N}}\frac{n!}{a_{1}!\cdots a_{N}!}\prod_{k=1}^{N}H^{v_{k}}_{a_{k}}(u_{k}).\tag{A}$$

Let $K$ be the set of $N\in\mathbb{N}$ such that (A) holds for all $v_{1},\dots,v_{N}\ge0$, $u_{1},\dots,u_{N}\in\mathbb{R}$ and $n\in\mathbb{N}_{0}$. Then $1\in K$: $T_{n,1}=\{(n)\}$, the coefficient is $n!/n!=1$, a sum over a singleton is its term (claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set), and sums and products with one term are that term (claim 1 of Properties of Finite Sums and of Properties of Finite Products). Let $N\in K$, let $v_{1},\dots,v_{N+1}\ge0$, $u_{1},\dots,u_{N+1}\in\mathbb{R}$ and $n\in\mathbb{N}_{0}$, and let $V=\sum_{k\le N}v_{k}\ge0$, $U=\sum_{k\le N}u_{k}$, so that $\sum_{k\le N+1}v_{k}=V+v_{N+1}$ and $\sum_{k\le N+1}u_{k}=U+u_{N+1}$ by claim 1 of Properties of Finite Sums. By Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §addition (with $v=V$, $w=v_{N+1}$, $s=U$, $t=u_{N+1}$), (A) for $N$ applied to each $H^{V}_{j}(U)$ (as $N\in K$), and claim 3 of Properties of Finite Sums (applied to the inner sums written along enumerations as in Sum over a Finite Index Set) to bring the factors $\binom{n}{j}H^{v_{N+1}}_{n-j}(u_{N+1})$ inside,

$$H^{V+v_{N+1}}_{n}(U+u_{N+1})=\sum_{j=0}^{n}\ \sum_{a\in T_{j,N}}\binom{n}{j}\frac{j!}{a_{1}!\cdots a_{N}!}\prod_{k=1}^{N}H^{v_{k}}_{a_{k}}(u_{k})\,H^{v_{N+1}}_{n-j}(u_{N+1}).$$

For $j\in\{0,\dots,n\}$ and $a\in T_{j,N}$ put $\varphi(j,a)=b=(a_{1},\dots,a_{N},n-j)$; then $b\in T_{n,N+1}$ by claim 1 of Properties of Finite Sums. By the definition of $\binom{n}{j}$ in Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers, $\binom{n}{j}\frac{j!}{a_{1}!\cdots a_{N}!}=\frac{n!}{a_{1}!\cdots a_{N}!\,(n-j)!}$, and by claim 1 of Properties of Finite Products (applied to the factorials and to the Hermite factors) the term indexed by $(j,a)$ equals $\frac{n!}{b_{1}!\cdots b_{N+1}!}\prod_{k=1}^{N+1}H^{v_{k}}_{b_{k}}(u_{k})$. Let $P$ be the set of ordered pairs $(j,a)$ with $j\in\{0,\dots,n\}$ and $a\in T_{j,N}$. The map $\varphi$ is a bijection from $P$ onto $T_{n,N+1}$, with inverse $b\mapsto(n-b_{N+1},(b_{1},\dots,b_{N}))$ (here $b_{N+1}\le n$ by claim 6 and $\sum_{k\le N}b_{k}=n-b_{N+1}$ by claim 1 of Properties of Finite Sums). The outer sum from $0$ (Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers) is the sum over the finite set $\{0,\dots,n\}$ in the sense of Sum over a Finite Index Set, computed along the enumeration $k\mapsto k-1$ of $[n+1]$: this is immediate if $n=0$, and follows from Splitting a Finite Sum at an Index (with $1$ and $n$ in place of $m$ and $n$ there) if $n\ge1$. Since each $T_{j,N}$ is nonempty and finite, claim 4 of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus turns the double sum into the sum over $P$ of the same terms. Finally, if $\chi:[p]\to P$ is a bijection, so is $\varphi\circ\chi:[p]\to T_{n,N+1}$, and by Sum over a Finite Index Set both the sum over $P$ and the sum over $T_{n,N+1}$ of the corresponding terms equal $\sum_{k=1}^{p}$ of the term indexed by $\chi(k)$, the choice of enumeration being immaterial since two enumerations differ by a permutation of $[p]$ (Invariance of Finite Sums and Products under Reindexing by a Permutation). This gives (A) for $N+1$ and the given data; so $N+1\in K$, and $K=\mathbb{N}$ by Principle of Induction for the Natural Numbers.

Step 2 (Claim 2). Let $h$ and $N$ be as in claim 2 and $x\in X$. By Step 0, $\ell_{h}(x)=\sum_{k=1}^{N}u_{k}$ with $u_{k}=(h_{k}/c_{k})x_{k}$, and $|h|_{c}^{2}=\sum_{k=1}^{N}v_{k}$ with $v_{k}=h_{k}^{2}/c_{k}\ge0$. By The Wick Powers of a Paley-Wiener Functional §wick and (A),

$${:}\ell_{h}^{n}{:}(x)=\sum_{a\in T_{n,N}}\frac{n!}{a_{1}!\cdots a_{N}!}\prod_{k=1}^{N}H^{v_{k}}_{a_{k}}(u_{k}).$$

By Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §scaling with $r=h_{k}/c_{k}$, $v=c_{k}$, $t=x_{k}$, and since $r^{2}c_{k}=h_{k}^{2}/c_{k}=v_{k}$, we get $H^{v_{k}}_{a_{k}}(u_{k})=(h_{k}/c_{k})^{a_{k}}H^{c_{k}}_{a_{k}}(x_{k})$. The map sending $a\in T_{n,N}$ to the multi-index with terms $a_{1},\dots,a_{N}$ followed by zeros is a bijection onto $\mathcal{A}_{n,N}$ ($N$ being a length bound for each element of $\mathcal{A}_{n,N}$, whose order is then $\sum_{k\le N}\alpha_{k}$); and with the length bound $N$, $\alpha!=\prod_{k\le N}a_{k}!$, $(h/c)^{\alpha}=\prod_{k\le N}(h_{k}/c_{k})^{a_{k}}$ (Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order) and $H_{\alpha}(x)=\prod_{k\le N}H^{c_{k}}_{a_{k}}(x_{k})$ (The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space §hermite). Substituting, with claim 2 of Properties of Finite Products to write $\prod_{k\le N}(h_{k}/c_{k})^{a_{k}}H^{c_{k}}_{a_{k}}(x_{k})$ as $(h/c)^{\alpha}H_{\alpha}(x)$, and reindexing the sum along this bijection as at the end of Step 1, gives claim 2; in particular $\mathcal{A}_{n,N}$, the image of the finite set $T_{n,N}$, is finite (claim 4 of Basic Properties of Finite Sets).

Step 3 (Covariance for finitely many coordinates). Let $h,g\in X$ and $N\in\mathbb{N}$ with $h_{k}=g_{k}=0$ for $k>N$ (so $h,g\in H_{c}$ by Step 0), and $m,n\in\mathbb{N}_{0}$. By claim 2, ${:}\ell_{h}^{m}{:}=\sum_{\alpha\in\mathcal{A}_{m,N}}a_{\alpha}H_{\alpha}$ and ${:}\ell_{g}^{n}{:}=\sum_{\beta\in\mathcal{A}_{n,N}}b_{\beta}H_{\beta}$ pointwise, with $a_{\alpha}=\frac{m!}{\alpha!}(h/c)^{\alpha}$ and $b_{\beta}=\frac{n!}{\beta!}(g/c)^{\beta}$. Each $H_{\alpha}$ is Borel with class in $L^{2}$ (Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical), so these Wick powers are Borel (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and $2$-integrable, their classes being the corresponding linear combinations of classes (The Lebesgue Space of Power-Integrable Functions §space). By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product the product ${:}\ell_{h}^{m}{:}\,{:}\ell_{g}^{n}{:}$ is integrable with integral $\langle{:}\ell_{h}^{m}{:},{:}\ell_{g}^{n}{:}\rangle_{L^{2}(\gamma_{c})}$, and by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations (twice; the sums over $\mathcal{A}_{m,N}$ and $\mathcal{A}_{n,N}$ are converted into finite sums by enumeration as in Sum over a Finite Index Set) and Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §orthogonality,

$$\int_{X}{:}\ell_{h}^{m}{:}\,{:}\ell_{g}^{n}{:}\,d\gamma_{c}=\sum_{\alpha\in\mathcal{A}_{m,N}}\sum_{\beta\in\mathcal{A}_{n,N}}a_{\alpha}b_{\beta}\langle H_{\alpha},H_{\beta}\rangle_{L^{2}(\gamma_{c})}.$$

If $m\ne n$, every pair has $|\alpha|\ne|\beta|$, so $\alpha\ne\beta$, every term is $0$, and so is the double sum, by claim 3 of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus. If $m=n$, then for fixed $\alpha$ every term of the inner sum with $\beta\ne\alpha$ vanishes, so by claim 3 of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus (with the subset $\{\alpha\}$) and claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set the inner sum is $a_{\alpha}b_{\alpha}\,\alpha!\,c^{\alpha}$. Moreover $(h_{k}/c_{k})^{a}(g_{k}/c_{k})^{a}c_{k}^{a}=(h_{k}g_{k}/c_{k})^{a}$ for $a\in\mathbb{N}_{0}$, by claim 3 of Properties of Natural Number Powers in a Field (trivially if $a=0$), so claim 2 of Properties of Finite Products, with the length bound $N$, gives $(h/c)^{\alpha}(g/c)^{\alpha}c^{\alpha}=\prod_{k=1}^{N}(h_{k}g_{k}/c_{k})^{\alpha_{k}}$. Hence

$$\int_{X}{:}\ell_{h}^{n}{:}\,{:}\ell_{g}^{n}{:}\,d\gamma_{c}=\sum_{\alpha\in\mathcal{A}_{n,N}}\Bigl(\frac{n!}{\alpha!}\Bigr)^{2}(h/c)^{\alpha}(g/c)^{\alpha}\,\alpha!\,c^{\alpha}=n!\sum_{a\in T_{n,N}}\frac{n!}{a_{1}!\cdots a_{N}!}\prod_{k=1}^{N}\Bigl(\frac{h_{k}g_{k}}{c_{k}}\Bigr)^{a_{k}}=n!\Bigl(\sum_{k=1}^{N}\frac{h_{k}g_{k}}{c_{k}}\Bigr)^{n},$$

using the bijection of Step 2 and Multinomial Theorem (with $m=N$, $d=n$, $a_{k}=h_{k}g_{k}/c_{k}$). By Step 0 the last expression is $n!\,\langle h,g\rangle_{c}^{\,n}$.

Step 4 (The truncated Wick powers form a Cauchy sequence). Let $h\in H_{c}$, $n\in\mathbb{N}_{0}$, and for $N\in\mathbb{N}$ let $W_{N}={:}\ell_{h^{[N]}}^{n}{:}$, an element of $L^{2}$ by Step 3, and $s_{N}=\sum_{k=1}^{N}h_{k}^{2}/c_{k}$. By Step 0, $|h^{[N]}|_{c}^{2}=s_{N}$ and, for $N\le M$, $\langle h^{[N]},h^{[M]}\rangle_{c}=\langle h^{[M]},h^{[N]}\rangle_{c}=s_{N}$. Let $N\le M$. Step 3, applied with the common bound $M$ to the three pairs $(h^{[M]},h^{[M]})$, $(h^{[M]},h^{[N]})$ and $(h^{[N]},h^{[N]})$, all of which vanish beyond $M$, together with The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, which identifies each integral computed there with the corresponding inner product in $L^{2}$, gives $\lVert W_{M}\rVert_{2}^{2}=n!\,s_{M}^{n}$, $\langle W_{M},W_{N}\rangle_{L^{2}(\gamma_{c})}=n!\,s_{N}^{n}$ and $\lVert W_{N}\rVert_{2}^{2}=n!\,s_{N}^{n}$. Hence, by Elementary Identities in a Real Inner Product Space §expansion,

$$\lVert W_{M}-W_{N}\rVert_{2}^{2}=n!\bigl(s_{M}^{n}-2s_{N}^{n}+s_{N}^{n}\bigr)=n!\bigl(s_{M}^{n}-s_{N}^{n}\bigr).$$

By The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §square and Series of Real Numbers §convergent, $s_{N}\to|h|_{c}^{2}$, so $s_{N}^{n}\to|h|_{c}^{2n}$ by the limit laws. Given $\varepsilon>0$ choose $N_{0}$ with $|s_{N}^{n}-|h|_{c}^{2n}|<\varepsilon^{2}/(2\,n!)$ for $N\ge N_{0}$; then $\lVert W_{M}-W_{N}\rVert_{2}^{2}<\varepsilon^{2}$, hence $\lVert W_{M}-W_{N}\rVert_{2}<\varepsilon$, for $M,N\ge N_{0}$. So $(W_{N})$ is Cauchy in $(L^{2},d)$, and since $L^{2}$ is complete (Real Hilbert Space §hilbert) it converges to some $W\in L^{2}$; let $w$ be a $2$-integrable representative of $W$.

Step 5 (Claim 1, and convergence of the truncations). Keep the notation of Step 4. Let $C$ be the set of $x\in X$ for which $(\ell_{h,N}(x))_{N}$ converges; it is Borel with $\gamma_{c}(C)=1$ by The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel and The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §almost-everywhere, and $\ell_{h,N}(x)\to\ell_{h}(x)$ for $x\in C$ by The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §functional. By Hermite Polynomials with a Given Variance §hermite, $H^{v}_{n}(t)$ is a finite sum of fixed real multiples of products $(-v)^{j}t^{n-2j}$, so the limit laws show: if $v_{N}\to v$ and $t_{N}\to t$ with all $v_{N},v\ge0$, then $H^{v_{N}}_{n}(t_{N})\to H^{v}_{n}(t)$. Since $W_{N}(x)=H^{s_{N}}_{n}(\ell_{h,N}(x))$ by Step 0, we get $W_{N}(x)\to{:}\ell_{h}^{n}{:}(x)$ for every $x\in C$. The function $H^{|h|_{c}^{2}}_{n}$ is of class $C^{2}$ on $\mathbb{R}^{1}$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §polynomial, so by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous it is continuous from $\mathbb{R}^{1}$ into $(\mathbb{R},d_{\mathbb{R}})$; identifying $\mathbb{R}^{1}$ with $\mathbb{R}$, whose Euclidean distance is $d_{\mathbb{R}}$ by The Euclidean Distance on the Real Line is the Absolute Value Metric, it is continuous from $(\mathbb{R},d_{\mathbb{R}})$ to itself, hence Borel by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space, and $\ell_{h}$ is Borel by The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel; so ${:}\ell_{h}^{n}{:}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Put $f_{N}=(W_{N}-w)^{2}$ and $u=({:}\ell_{h}^{n}{:}-w)^{2}$, nonnegative measurable functions (Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). We use the following fact (L): if $(r_{N})_{N\in\mathbb{N}}$ is a sequence of nonnegative reals converging to a real $r$, then $\liminf_{N}r_{N}=\sup_{K}\inf_{N\ge K}r_{N}$, formed in $[0,\infty]$ as in Fatou's Lemma, equals $r$. Indeed, for $K\in\mathbb{N}$ the infimum $\iota_{K}=\inf\{r_{N}:N\ge K\}$ exists in $[0,\infty]$ (Measure, Measure Space, and Probability Measure), and $\iota_{K}\le\iota_{K'}$ for $K\le K'$, the second set being contained in the first. Let $\varepsilon>0$ and, by Limit of a Sequence of Real Numbers, let $K_{\varepsilon}\in\mathbb{N}$ with $|r_{N}-r|<\varepsilon$ for all $N\ge K_{\varepsilon}$. For $K\ge K_{\varepsilon}$, $r-\varepsilon$ is a lower bound of $\{r_{N}:N\ge K\}$, so $r-\varepsilon\le\iota_{K}\le r_{K}<r+\varepsilon$; for $K<K_{\varepsilon}$, $\iota_{K}\le\iota_{K_{\varepsilon}}<r+\varepsilon$. Hence $r+\varepsilon$ is an upper bound of all $\iota_{K}$, and $r-\varepsilon\le\iota_{K_{\varepsilon}}\le\sup_{K}\iota_{K}\le r+\varepsilon$. So $\sigma=\sup_{K}\iota_{K}$ is real and $|\sigma-r|\le\varepsilon$ for every $\varepsilon>0$; if $\sigma\ne r$, the choice $\varepsilon=|\sigma-r|/2$ would give a contradiction, so $\sigma=r$. For $x\in C$, $f_{N}(x)\to u(x)$ by the limit laws, so $u=\liminf_{N}f_{N}$ on $C$ by (L), hence almost everywhere. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and Fatou's Lemma,

$$\int_{X}u\,d\gamma_{c}=\int_{X}\Bigl(\liminf_{N}f_{N}\Bigr)d\gamma_{c}\le\liminf_{N}\int_{X}f_{N}\,d\gamma_{c}=\liminf_{N}\lVert W_{N}-W\rVert_{2}^{2}=0,$$

where $\int_{X}f_{N}\,d\gamma_{c}=\lVert W_{N}-W\rVert_{2}^{2}$ by Power-Integrable Functions and the p-Seminorm §seminorm and The Lebesgue Space of Power-Integrable Functions §norm, and the last equality holds by (L), because $\lVert W_{N}-W\rVert_{2}^{2}\to0$ by the limit laws. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, $u=0$ almost everywhere, that is ${:}\ell_{h}^{n}{:}=w$ almost everywhere. Hence $({:}\ell_{h}^{n}{:})^{2}=w^{2}$ almost everywhere and $\int_{X}({:}\ell_{h}^{n}{:})^{2}\,d\gamma_{c}=\int_{X}w^{2}\,d\gamma_{c}<\infty$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison. This proves claim 1; moreover the class of ${:}\ell_{h}^{n}{:}$ is $W$ (The Lebesgue Space of Power-Integrable Functions §equivalence), so

$$\bigl\lVert{:}\ell_{h^{[N]}}^{n}{:}-{:}\ell_{h}^{n}{:}\bigr\rVert_{2}\to0\qquad(N\to\infty).\tag{T}$$

Step 6 (Claim 3). Let $h,g\in H_{c}$ and $m,n\in\mathbb{N}_{0}$. By claim 1 and The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, the product ${:}\ell_{h}^{m}{:}\,{:}\ell_{g}^{n}{:}$ is integrable with integral $\langle{:}\ell_{h}^{m}{:},{:}\ell_{g}^{n}{:}\rangle_{L^{2}(\gamma_{c})}$. By (T), which Steps 4 and 5 establish for every vector in $H_{c}$ and every exponent in $\mathbb{N}_{0}$, applied to $h$ with exponent $m$ and to $g$ with exponent $n$, and by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, $\langle{:}\ell_{h^{[N]}}^{m}{:},{:}\ell_{g^{[N]}}^{n}{:}\rangle_{L^{2}(\gamma_{c})}$ converges to this integral as $N\to\infty$. By Step 3 (with $h^{[N]},g^{[N]}$, which vanish beyond $N$), where each such inner product was identified with the integral computed there, these inner products are $0$ if $m\ne n$, and equal $n!\bigl(\sum_{k=1}^{N}h_{k}g_{k}/c_{k}\bigr)^{n}$ if $m=n$; the latter converges to $n!\,\langle h,g\rangle_{c}^{\,n}$ by The Cameron-Martin Inner Product of Two Cameron-Martin Vectors §inner-product, Series of Real Numbers §convergent and the limit laws. Uniqueness of limits gives claim 3.

Step 7 (Claim 4). For $f\in H_{c}$, $\langle f,f\rangle_{c}=|f|_{c}^{2}$, the two series being the same. Let $h^{(j)}$ be as in claim 4, $d^{(j)}=h^{(j)}-h\in H_{c}$ and $\delta_{j}=|d^{(j)}|_{c}^{2}\to0$. By Elementary Identities in a Real Inner Product Space §bilinear and symmetry, $h^{(j)}_{k}=h_{k}+d^{(j)}_{k}$. By claim 3, applied to the three pairs $(h^{(j)},h^{(j)})$, $(h^{(j)},h)$ and $(h,h)$ with $m=n$, and The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, which identifies each integral there with the corresponding inner product in $L^{2}$, we get $\lVert{:}\ell_{h^{(j)}}^{n}{:}\rVert_{2}^{2}=n!\,|h^{(j)}|_{c}^{2n}$, $\langle{:}\ell_{h^{(j)}}^{n}{:},{:}\ell_{h}^{n}{:}\rangle_{L^{2}(\gamma_{c})}=n!\,\langle h^{(j)},h\rangle_{c}^{\,n}$ and $\lVert{:}\ell_{h}^{n}{:}\rVert_{2}^{2}=n!\,|h|_{c}^{2n}$; so Elementary Identities in a Real Inner Product Space §expansion gives

$$\bigl\lVert{:}\ell_{h^{(j)}}^{n}{:}-{:}\ell_{h}^{n}{:}\bigr\rVert_{2}^{2}=n!\Bigl(|h^{(j)}|_{c}^{2n}-2\langle h^{(j)},h\rangle_{c}^{\,n}+|h|_{c}^{2n}\Bigr).$$

Termwise, $h^{(j)}_{k}h_{k}/c_{k}=h_{k}^{2}/c_{k}+d^{(j)}_{k}h_{k}/c_{k}$ and $(h^{(j)}_{k})^{2}/c_{k}=h_{k}^{2}/c_{k}+2d^{(j)}_{k}h_{k}/c_{k}+(d^{(j)}_{k})^{2}/c_{k}$, so Elementary Properties of Series of Real Numbers §linearity gives $\langle h^{(j)},h\rangle_{c}=|h|_{c}^{2}+\langle d^{(j)},h\rangle_{c}$ and $|h^{(j)}|_{c}^{2}=|h|_{c}^{2}+2\langle d^{(j)},h\rangle_{c}+\delta_{j}$. Let $t>0$. For every $k$, $0\le(t|d^{(j)}_{k}|-|h_{k}|)^{2}$ gives $|d^{(j)}_{k}h_{k}|/c_{k}\le\tfrac12\bigl(t(d^{(j)}_{k})^{2}/c_{k}+h_{k}^{2}/(tc_{k})\bigr)$; the series of the right-hand sides converges with sum $\tfrac12(t\delta_{j}+|h|_{c}^{2}/t)$ by Elementary Properties of Series of Real Numbers §linearity, so An Absolutely Convergent Series of Real Numbers Converges §dominated gives $|\langle d^{(j)},h\rangle_{c}|\le\tfrac12(t\delta_{j}+|h|_{c}^{2}/t)$. Given $\varepsilon>0$, take $t=(|h|_{c}^{2}+1)/\varepsilon$, so that $|h|_{c}^{2}/(2t)<\varepsilon/2$, and $j_{0}$ with $\delta_{j}<\varepsilon/t$ for $j\ge j_{0}$; then $|\langle d^{(j)},h\rangle_{c}|<\varepsilon$ for $j\ge j_{0}$. Hence $\langle d^{(j)},h\rangle_{c}\to0$, and by the limit laws $\langle h^{(j)},h\rangle_{c}\to|h|_{c}^{2}$, $|h^{(j)}|_{c}^{2}\to|h|_{c}^{2}$, and the right side of the display converges to $n!(|h|_{c}^{2n}-2|h|_{c}^{2n}+|h|_{c}^{2n})=0$. Since the norms are nonnegative and a nonnegative real is below $\varepsilon$ when its square is below $\varepsilon^{2}$, the norms converge to $0$. This proves claim 4.

Step 8 (Claim 5). For $N\in\mathbb{N}$, $h^{[N]}_{k}=0$ for $k>N$ (Step 0), so by claim 2 the function ${:}\ell_{h^{[N]}}^{n}{:}$ is the linear combination $\sum_{\alpha\in\mathcal{A}_{n,N}}\frac{n!}{\alpha!}(h^{[N]}/c)^{\alpha}H_{\alpha}$ over the nonempty finite set $\mathcal{A}_{n,N}\subseteq\mathcal{A}_{n}$ (it contains the multi-index with first term $n$ and all others $0$). By The Lebesgue Space of Power-Integrable Functions §space its class is the same combination of the classes $H_{\alpha}$, an element of the linear span of $\{H_{\alpha}:\alpha\in\mathcal{A}_{n}\}$ in the sense of The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space. By (T) and Sequential Characterization of the Closure in a Metric Space, ${:}\ell_{h}^{n}{:}$ lies in the closure of that span, which is $\mathcal{H}_{n}$ by The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space §chaos.