Adjacent iid vector functionals admit a finite-moment strong approximation

Write $a(x)=\int h(x,y)\nu (dy)$, $b(y)=\int h(x,y)\nu (dx)$ and $c(x,y)=h(x,y)-a(x)-b(y)$, defining them arbitrarily on the null sets where the integrals are not finite. By [F4], they are measurable, have finite $q$th moments, are centered, and $\mathbb{E}[c(U,V)\mid U]=\mathbb{E}[c(U,V)\mid V]=0$ for independent $U,V$ of law $\nu$. Expanding the products using these two zero conditional means gives $\mathrm{\Gamma }=\mathrm{cov}(a(U)+b(U))+\mathrm{cov}(c(U,V))\ge 0.$ Indeed the variance of $h(U,V)$ is the sum of the covariances of $a(U)$, $b(V)$ and $c(U,V)$, while the lag-one covariance is $\mathbb{E}[b(U)a(U{)}^T]$. This proves both the displayed identity and nonnegativity without any limit theorem.

We derive the finite-horizon iid bound needed for this changing row directly from [F2]. Suppose $T_1,\dots ,T_N$ are iid centered vectors with $\mathbb{E}\mathrm{\mid }{T}_1{\mathrm{\mid }}^q\le M$, and covariance $\mathrm{\Sigma }$ whose eigenvalues lie in a fixed $[\lambda ,\mathrm{\Lambda }]$ with $\lambda >0$. Put $R_2=N^{1\mathrm{/}\beta }$ and $T_i^{\mathrm{\prime }}=T_i{\mathbf{1}}_{\{\mathrm{\mid }{T}_i\mathrm{\mid }\le R_2\}}-\mu$, $\mu =\mathbb{E}[T_1{\mathbf{1}}_{\{\mathrm{\mid }{T}_1\mathrm{\mid }\le R_2\}}]$. The tail union probability is at most $MN{R}_2^{-q}$, and on its complement the maximum sum correction is at most $MN{R}_2^{1-q}=O(R_2)$. Also $\mathrm{\Sigma }-{\mathrm{\Sigma }}^{\mathrm{\prime }}=\mathbb{E}[T_1{T}_1^T{\mathbf{1}}_{\{\mathrm{\mid }{T}_1\mathrm{\mid }>R_2\}}]+\mu {\mu }^T$, so $\mathrm{\parallel }\mathrm{\Sigma }-{\mathrm{\Sigma }}^{\mathrm{\prime }}\mathrm{\parallel }\le C_M{R}_2^{2-q}$. Uniformly for all sufficiently large $N$ the eigenvalues of ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ are in $[\lambda \mathrm{/}2,C]$. The whitened $T_i^{\mathrm{\prime }}$ have covariance $I$ and norm at most $C{R}_2$, so [F2] yields independent standard $H_i$ with maximum coupling error to ${\mathrm{\Sigma }}^{\mathrm{\prime }1\mathrm{/}2}{H}_i$ at most $C{R}_2{\log }^3(eN)$, outside probability $N^{-A}$. All constants are uniform in the iid law under these moment and eigenvalue bounds.

For a deterministic orthonormal basis of $\ker \mathrm{\Gamma }$, step 1.1's sum of nonnegative scalar variances implies that each basis component of $c$ is zero almost surely and the same component of $a+b$ is zero almost surely. Hence the kernel projection of $h(U_i,U_{i+1})$ is $a_0(U_i)-a_0(U_{i+1})$, where $a_0$ is the kernel projection of $a$. It is an exact endpoint coboundary simultaneously for all $i$, after removing a countable union of null sets. Its partial sums are $a_0(U_0)-a_0(U_k)$. For every $ϵ>0$, the union and Markov bounds over $k\le 2^j$ give probability at most $C{ϵ}^{-q}{2}^{j(1-q\mathrm{/}p)}$ that their maximum exceeds $ϵ2^{j\mathrm{/}p}$. This is summable, so [F7], first for rational $ϵ>0$ and then for all $n$ between successive powers of two, gives maximum $o(n^{1\mathrm{/}p})$. If $\mathrm{\Gamma }=0$ take $G_i=0$ and adjoin a Brownian motion by [F8]. Otherwise work in its deterministic positive-covariance range of dimension one or two and later restore this kernel projection.

For independent anchors $A,B$ and interior variables $V,W$, all of law $\nu$, put $Y=h(A,V)-a(A)+h(V,W)+h(W,B)-b(B)$. Its conditional mean given $A,B$ is zero. Its average conditional covariance is ${\mathrm{\Gamma }}_{\ast }:=\mathbb{E}\mathrm{cov}(Y\mid A,B)=2\mathrm{cov}(a(U)+b(U))+3\mathrm{cov}(c(U,V))\mathrm{.}$ To check this, substitute the decomposition from step 1.1 to write $Y=(a+b)(V)+(a+b)(W)+c(A,V)+c(V,W)+c(W,B)$. Every cross product vanishes after integrating an endpoint using the zero conditional means of $c$; the five individual covariance contributions give the formula. Thus $2\mathrm{\Gamma }\le {\mathrm{\Gamma }}_{\ast }\le 3\mathrm{\Gamma }$, so ${\mathrm{\Gamma }}_{\ast }$ is positive definite on the range retained in step 2.1. A fixed invertible linear map there makes ${\mathrm{\Gamma }}_{\ast }=I$; its inverse will be restored at the end, and all subsequent constants may depend on this fixed map.

Choose fixed $p<\beta <\alpha <q$, and consider an epoch of $N=2^j$ consecutive triple indices. Put $R=N^{1\mathrm{/}\alpha }$, $L=K\log (eN)$ and $m=\lceil R^{2}L\rceil$. In this epoch replace the normalized active $h$ by $f=h{\mathbf{1}}_{\{\mathrm{\mid }h\mathrm{\mid }\le R\}}-\mathbb{E}[h{\mathbf{1}}_{\{\mathrm{\mid }h\mathrm{\mid }\le R\}}]$. Then $\mathrm{\mid }f\mathrm{\mid }\le 2R$, $\mathbb{E}\mathrm{\mid }f{\mathrm{\mid }}^q\le C$, and its means $a_R,b_R$ and degenerate component $c_R$ have uniformly bounded $q$th moments by [F4]. They converge in $L^2$ to their untruncated versions: the squared tail is at most $R^{2-q}\mathbb{E}\mathrm{\mid }h{\mathrm{\mid }}^q$, and conditional averaging is an $L^2$ contraction by coordinatewise [F4]. Consequently the average conditional triple-noise covariance ${\mathrm{\Gamma }}_{\ast ,R}=2\mathrm{cov}(a_R+b_R)+3\mathrm{cov}(c_R)$ tends to $I$. For all sufficiently large deterministic epochs, $c_{0}I\le {\mathrm{\Gamma }}_{\ast ,R}\le C_{0}I$ for fixed $c_0,C_0>0$. Finitely many earlier epochs can be coupled arbitrarily and contribute only finite errors.

Let $A_t=U_{3t}$, $V_t=U_{3t+1}$ and $W_t=U_{3t+2}$ in the epoch, relabeling its first triple index as zero. Conditional on its anchors $(A_0,\dots ,A_N)$, the interior pairs $(V_t,W_t)$ are independent, and the centered triple noises $Y_t=f(A_t,V_t)-a_R(A_t)+f(V_t,W_t)+f(W_t,A_{t+1})-b_R(A_{t+1})$ are independent centered vectors bounded by $CR$. Define the symmetric matrices $Q(z)=\mathrm{cov}(f(V,W)+f(W,z)-b_R(z))$ and $P(z)=\mathrm{cov}(f(z,V)-a_R(z))+\mathrm{cov}(f(z,V),f(V,W))+\mathrm{cov}(f(V,W),f(z,V))$, where these covariances integrate independent $V,W$. The left and right endpoint terms are independent given the anchors, so $V_t:=\mathrm{cov}(Y_t\mid A_t,A_{t+1})=P(A_t)+Q(A_{t+1}),D_t:=P(A_t)+Q(A_t)=\mathrm{cov}(Y(A_t,A_t)\mid A_t)\mathrm{.}$ Although $P$ need not be positive semidefinite, $V_t,D_t,Q(A_t)$ are positive semidefinite, their norms are at most $C{R}^2$, and $\mathbb{E}{V}_t=\mathbb{E}{D}_t={\mathrm{\Gamma }}_{\ast ,R}$. Moreover, for every interval of triple indices, ${\sum }_{t=u}^{v-1}(V_t-D_t)=Q(A_v)-Q(A_u)\mathrm{.}$ These are identities of measurable anchor functions, by [F4] and [F6].

Divide the epoch into consecutive chunks of length $m$, leaving a shorter final chunk. The entries of $V_t$ are one-dependent functions of adjacent iid anchors, while $D_t$ are iid functions of individual anchors. For a positive semidefinite matrix $T$ of norm at most $C{R}^2$, $\mathrm{\mid }{T}_{uv}{\mathrm{\mid }}^2\le T_{uu}{T}_{vv}\le C{R}^2{T}_{uu}$. Its entrywise second moments here are therefore at most $C{R}^2$, because the mean matrices in step 5.1 are uniformly bounded. Within each parity the entries of $V_t$ are independent. Apply [F3] separately to both parities and each matrix entry, and to the independent entries of $D_t$ across $t$. For any full chunk $I$, the probability that an entry of either centered covariance sum exceeds a fixed small multiple of $m$ is at most $C\exp (-cm\mathrm{/}{R}^2)$. Since $\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le d{\max }_{u,v}\mathrm{\mid }{T}_{uv}\mathrm{\mid }$ for symmetric matrices in dimension $d\le 2$, choose the multiplier small relative to $c_0$ and union over at most $N$ chunks and their entries. Choosing $K$ sufficiently large gives an anchor event of probability at least $1-N^{-A}$, for any prescribed $A>0$, on which $c_{1}mI\le {\sum }_{t\in I}{V}_t\le C_{1}mI,c_{1}mI\le {\sum }_{t\in I}{D}_t\le C_{1}mI$ for every full chunk, with both sums bounded on both sides. The same entrywise Bernstein calculation gives an upper bound $C_{1}mI$ on both sums in the shorter chunk. Here and below the constants may be enlarged independently of $N$ and $R$. Since $\alpha >2$, $m<N$ eventually; if not, the same upper bound handles the entire epoch as one residual chunk.

On a good anchor vector, divide each full-chunk sum of $Y_t$ by $\sqrt{c_{1}m}$. Its conditional covariance lies between $I$ and $KI$, for a fixed $K$, and it is a sum of independent centered atoms of norm at most $CR\mathrm{/}\sqrt{m}\le C$. Its whole value need not be bounded. Its atom kernels are the measurable pushforwards of $\nu \otimes \nu$ under the formula in step 5.1; independence holds conditional on the supplied anchors. Thus [F1], with the ambient length parameter $N$ and a fixed atom bound $B\ge \max (1,C)$, supplies a measurable coupling of the chunk sums with independent matching Gaussian chunk sums. Undoing the normalization gives maximum chunk-boundary error at most $C\sqrt{m}{\log }^3(eN)\le CR{\log }^{7\mathrm{/}2}(eN)$, with conditional failure at most $N^{-A}$. On bad anchors use independent product coupling instead, preserving both marginals. No theorem about uniformly bounded grouped values is being invoked.

To redistribute covariance without a matrix random clock, start instead with independent standard Gaussian vectors $Z_t$, independent of the anchors, and put $Z_t^D=D_t^{1\mathrm{/}2}{Z}_t$. These have the required conditional independent $D_t$ row law even if individual $D_t$ are singular. On good anchors group them into the same full chunks, whose covariances $B_i=\sum D_t$ are positive definite. Write $A_i=\sum V_t$ for the desired other row. Pad to a dyadic number of chunks with independent identical-covariance $mI$ Gaussian entries in both rows. All chunk covariances lie between $c_{1}mI$ and $C_{1}mI$, and every contiguous whole-chunk interval satisfies $\mathrm{\parallel }{A}_I-B_I\mathrm{\parallel }\le C{R}^2$ by step 5.1's endpoint identity; padded intervals retain that bound. We construct the $A$ row from the already given $B$ row using the Gaussian tree in the next two steps, leaving the original $Z_t$ unchanged.

Fill a Gaussian chunk of total covariance $S>0$ and coupled sum $z$ with increments of individual covariances $V_t$ by taking independent $H_t\sim N(0,V_t)$ and setting $H_t^{\mathrm{\prime }}=H_t+V_t{S}^{-1}(z-\sum H_t)$. The residual $H_t-V_t{S}^{-1}\sum H_t$ is independent of the original total Gaussian by [F5]; direct covariance calculation shows this is its exact conditional bridge law and its sum is $z$. Thus the completed Gaussian noise row has exactly its conditional independent Gaussian product law. Conditional on good anchors, every within-chunk raw prefix has atom bound $CR$ and covariance at most $C_{1}mI$. Coordinatewise [F3] and Gaussian tails in [F5], with a union bound over at most $N$ prefixes, bound both raw and Gaussian within-chunk norms by $C(\sqrt{mL}+RL)\le CRL$, with failure at most $N^{-A}$ after enlarging $K$. Couple a shorter residual chunk independently and use its covariance upper bound in the same calculation. Consequently there is a conditional noise-row coupling $Y_t\leftrightarrow Z_t^V$, with $Z_t^V$ conditionally independent $N(0,V_t)$, of full-prefix error $CR{\log }^{7\mathrm{/}2}(eN)$ and total failure $C{N}^{-A}$. Its extension to the actually given $Y_t$ follows by [F6] disintegrating the finite joint law given anchors and $Y_t$; the original innovations are preserved.

At a node of $s$ chunks and for either covariance row $C$, write $C_P=C_L+C_R$, $H_C=C_L{C}_P^{-1}$, and $T_C=C_L-C_L{C}_P^{-1}{C}_L=(C_L^{-1}+C_R^{-1}{)}^{-1}$. Thus $C_P$ and $T_C$ have eigenvalues comparable to $sm$, and the Gaussian parent and residual $S_L-H_C{S}_P$ are independent by [F5]. Extract their standardized independent Gaussian primitives from the given $B$ row, and use the same primitives to construct the $A$ root and residuals. For positive definite matrices $M,N$, the identity $M^{1\mathrm{/}2}X+X{N}^{1\mathrm{/}2}=M-N$, with $X=M^{1\mathrm{/}2}-N^{1\mathrm{/}2}$, yields $\mathrm{\parallel }X\mathrm{\parallel }\le C\mathrm{\parallel }M-N\mathrm{\parallel }\mathrm{/}(\sqrt{{\lambda }_{\min }M}+\sqrt{{\lambda }_{\min }N})$: solve it by integrating $e^{-t{M}^{1\mathrm{/}2}}(M-N)e^{-t{N}^{1\mathrm{/}2}}$ and bound the integral. The endpoint bounds in step 7.2 and the inverse identity imply $\mathrm{\parallel }{H}_A-H_B\mathrm{\parallel }\le C{R}^2\mathrm{/}(sm)$, $\mathrm{\parallel }{T}_A-T_B\mathrm{\parallel }\le C{R}^2$, and $\mathrm{\parallel }{T}_A^{1\mathrm{/}2}-T_B^{1\mathrm{/}2}\mathrm{\parallel }\le C{R}^2\mathrm{/}\sqrt{sm}$. A union bound over the Gaussian parents and primitives gives, outside probability $N^{-A}$, $\mathrm{\mid }{S}_P^B\mathrm{\mid }\le C\sqrt{smL}$ and primitive norms at most $C\sqrt{L}$. Reconstruct children by $S_L^C=H_C{S}_P^C+T_C^{1\mathrm{/}2}{z}_I$ and $S_R^C=S_P^C-S_L^C$. Their errors satisfy $e_L=H_A{e}_P+{\eta }_I$, $e_R=(I-H_A)e_P-{\eta }_I$, with $\mathrm{\mid }{\eta }_I\mathrm{\mid }\le C{R}^2\sqrt{L\mathrm{/}(sm)}\le CR$, and the root error is also at most $CR$. Every child and root marginal is its exact Gaussian law, so the two completed chunk rows each have their required conditional product law.

The prefix bound uses exact covariance interpolation, not repeated norm bounds on $H_A$. In any subtree its leading root coefficient is $A_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{x}}{A}_P^{-1}$. At a boundary ancestor its residual coefficient is $A_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}{A}_L^{-1}$ if the boundary is in the left child, and $I-A_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{A}_R^{-1}$ if it is in the right child. For the left case substitute the child's leading term and cancel $A_L^{-1}{A}_L$; for the right case add the full left error to the partial right error, giving leading coefficient $(A_L+A_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}})A_P^{-1}$ and the stated residual coefficient. This proves the expansion by induction. Each coefficient has norm at most $1+C_1\mathrm{/}{c}_1$, using the count and eigenvalue bounds on chunk covariances, and there are at most $C\log (eN)$ boundary ancestors. Step 8.2 therefore gives prefix error $CR\log (eN)$ at chunk boundaries. Fill the $V_t$ Gaussian increments by the bridge in step 8.1; the $D_t$ increments were already present. Their within-chunk prefix norms are at most $C\sqrt{mL}\le CRL$ by Gaussian tails and the covariance upper bounds. Handle a short residual chunk independently. This gives full-prefix Gaussian-row error $CR\log (eN)$ outside probability $C{N}^{-A}$. On bad anchors use an independent $V_t$ row while retaining the original independent standard $Z_t$. All operations are measurable matrix functions and finite Gaussian kernels.

The couplings of steps 8.1 and 9.1 have the same joint anchor and $V_t$-Gaussian-row marginal. Disintegrate the second joint law to give its standard $Z$ row conditional on that common anchor and Gaussian row, and append it to the first coupling. The disintegration formula in [F6] proves that both joint laws, hence both error estimates, are preserved. In particular the original innovations remain unchanged and the $Z_t$ row is still independent standard Gaussian conditional on anchors. The two full-prefix errors add. Adding back the conditional means of the triples now gives an iid replacement row $L_t=D_t^{1\mathrm{/}2}{Z}_t+a_R(A_t)+b_R(A_t),$ since the pairs $(A_t,Z_t)$ are iid. Indeed the original mean sum is ${\sum }_{t<k}(a_R(A_t)+b_R(A_{t+1}))={\sum }_{t<k}(a_R(A_t)+b_R(A_t))+b_R(A_k)-b_R(A_0)$. The endpoint means have uniformly bounded $q$th moments by step 4.1. Their maximum exceeds $R$ with probability at most $CN{R}^{-q}$ by [F7]. Therefore the original truncated triple sums and the partial sums of $L_t$ have maximum error $CR{\log }^{7\mathrm{/}2}(eN)$ with failure at most $C{N}^{-A}+CN{R}^{-q}$.

The row $L_t$ is centered and has uniformly bounded $q$th moment: conditional Gaussian moments and coordinatewise [F4] give $\mathbb{E}\mathrm{\parallel }{D}_t{\mathrm{\parallel }}^{q\mathrm{/}2}\le C\mathbb{E}\mathrm{\mid }Y(A_t,A_t){\mathrm{\mid }}^q\le C\mathbb{E}\mathrm{\mid }f(U,V){\mathrm{\mid }}^q\le C$, and the mean terms satisfy the same bound. Write ${\gamma }_0=\mathbb{E}[f(U_0,U_1)f(U_0,U_1{)}^T]$, ${\gamma }_1=\mathbb{E}[f(U_0,U_1)f(U_1,U_2{)}^T]$, and ${\mathrm{\Gamma }}_R={\gamma }_0+{\gamma }_1+{\gamma }_1^T$. The variance of a triple is $3{\gamma }_0+2({\gamma }_1+{\gamma }_1^T)$. Subtracting the covariance of its conditional mean $a_R(A_0)+b_R(A_1)$ gives $\mathbb{E}{D}_t$ by total variance. Since $\mathbb{E}[b_R(U)a_R(U{)}^T]={\gamma }_1$, adding $\mathrm{cov}(a_R+b_R)$ gives $\mathrm{cov}(L_t)=3{\mathrm{\Gamma }}_R\mathrm{.}$ This is a direct finite product calculation, independent of convergence in law or an inference from a coupling error. As $R\to \mathrm{\infty }$, ${\mathrm{\Gamma }}_R\to \mathrm{\Gamma }$, which is positive definite in the active space; hence these iid covariances have fixed upper and positive lower eigenvalue bounds in every sufficiently large epoch.

Step 8.2's square-root bound gives $\mathrm{\parallel }{\mathrm{\Sigma }}^{1\mathrm{/}2}-{\mathrm{\Sigma }}^{\mathrm{\prime }1\mathrm{/}2}\mathrm{\parallel }\le C_M{R}_2^{2-q}$. Use the same $H_i$ and apply Gaussian tails to all its partial sums to bound the covariance correction by $C_M\sqrt{N}{R}_2^{2-q}\sqrt{\log (eN)}$ with failure at most $N^{-A}$. This is $o(R_2)$ because $\beta <q<2(q-1)$. Consequently step 1.2 and this covariance correction give a finite-row iid coupling with matching covariance $\mathrm{\Sigma }$, error $C{R}_2{\log }^3(eN)$ and failure $C{N}^{-A}+MN{R}_2^{-q}$. Apply this result to the varying laws $T_i=L_i$ from step 11.1, using their uniform moment and eigenvalue bounds. This uses no fixed-law asymptotic iid theorem and asserts no triangular-law interface absent from a supplier's Statement.

The original normalized active $h$ and $f$ agree up to their global mean correction unless one of the epoch's $3N$ observations exceeds $R$, an event of probability at most $CN{R}^{-q}$. The mean correction is at most $CN{R}^{1-q}=O(R)$. Moreover $\mathrm{\parallel }{\mathrm{\Gamma }}_R-\mathrm{\Gamma }\mathrm{\parallel }\le C{R}^{2-q}$. For the lag-one product difference, use $\mathrm{\mid }{h}_0{h}_1^T-f_0{f}_1^T\mathrm{\mid }\le C(\mathrm{\mid }{h}_0\mathrm{\mid }+\mathrm{\mid }{h}_1\mathrm{\mid }{)}^2{\mathbf{1}}_{\{\mathrm{\mid }{h}_0\mathrm{\mid }+\mathrm{\mid }{h}_1\mathrm{\mid }>R\}}$ before centering and then include the mean correction. Pointwise the displayed squared tail is at most a constant times the sum of the squared tails of $\mathrm{\mid }{h}_0\mathrm{\mid }$ and $\mathrm{\mid }{h}_1\mathrm{\mid }$ above $R\mathrm{/}2$, so its expectation is at most $C{R}^{2-q}\mathbb{E}\mathrm{\mid }h{\mathrm{\mid }}^q$. The variance term has the same bound. Since $\mathrm{\Gamma }$ is positive definite in the active space, couple the iid Gaussian triple increments of covariance $3{\mathrm{\Gamma }}_R$ from step 12.1 to increments of covariance $3\mathrm{\Gamma }$ using the same standard normals. Step 8.2's square-root bound and Gaussian tails make the maximum correction at most $C\sqrt{N}{R}^{2-q}\sqrt{\log (eN)}=o(R)$ with failure $N^{-A}$, because $\alpha <q<2(q-1)$. Thus each sufficiently large epoch of original triple sums can be coupled to independent Gaussian increments of covariance $3\mathrm{\Gamma }$, with error $C{N}^{1\mathrm{/}\beta }{\log }^{7\mathrm{/}2}(eN)$ and failure at most $C{N}^{-A}+C{N}^{1-q\mathrm{/}\alpha }+C{N}^{1-q\mathrm{/}\beta }\mathrm{.}$ Here $R=N^{1\mathrm{/}\alpha }\le N^{1\mathrm{/}\beta }$, and all constants are uniform over epochs.

Retain triple index $t=0$ as a finite initial prefix, coupled arbitrarily, and perform the finite constructions on epochs $\{2^j,\dots ,2^{j+1}-1\}$ of triple indices, $j\ge 0$, using fresh kernels. Each conditional noise coupling and each Gaussian redistribution has the full conditional product marginal given all anchors. Because the original interior pairs are independent conditional on anchors, their finite-epoch kernel products give a product standard $Z$ law conditional on all anchors. Hence the pairs $(A_t,Z_t)$, and therefore the replacement variables $L_t$ with the deterministic epoch-dependent function in step 10.1, are independent across epochs as well as within them. Each later iid-row coupling is glued through its finite replacement row by [F6]; because these rows are independent, products of their kernels give independent final Gaussian rows across epochs. These statements follow by integrating finite rectangles under the conditional product laws, so no independence is being inferred from an error bound. All anchor sets and row coordinates involved in an epoch are finite standard-Borel products, making these disintegrations legitimate. Apply [F6]'s countable kernel extension with the entire original probability space as its initial coordinate to preserve all original innovations. The failure bounds in step 13.1 are summable over $N=2^j$, since $\beta <\alpha <q$. By [F7], all sufficiently large epochs satisfy their error bounds almost surely; finitely many earlier errors can be included in a finite random constant.

Sum the completed epoch errors and the current partial one. The geometric bound ${\sum }_{j\le J}{2}^{j\mathrm{/}\beta }(j+1{)}^{7\mathrm{/}2}\le C_{\beta }{2}^{J\mathrm{/}\beta }(J+1{)}^{7\mathrm{/}2}$ gives a global triple-prefix error $O(n^{1\mathrm{/}\beta }{\log }^{7\mathrm{/}2}(en))=o(n^{1\mathrm{/}p})$. Map back from the fixed normalization in step 3.1 and restore the covariance-kernel coboundary from step 2.1. Split each independent Gaussian triple increment $H_t\sim N(0,3\mathrm{\Gamma })$ into three increments of covariance $\mathrm{\Gamma }$: for each $t$ take a fresh triple $K_{t,0},K_{t,1},K_{t,2}\sim N(0,\mathrm{\Gamma })$, independent across all $t,r$ and of all triple increments, and put $G_{3t+r}=H_t\mathrm{/}3+K_{t,r}-(K_{t,0}+K_{t,1}+K_{t,2})\mathrm{/}3$. By [F5] these three increments are jointly Gaussian, each has covariance $\mathrm{\Gamma }$ and distinct ones have zero cross covariance, so the entire resulting row is independent with covariance $\mathrm{\Gamma }$. At nontriple endpoints the original sum changes by at most two $X_i$, and the Gaussian sum by at most two $G_i$. Union and Markov bounds on dyadic ranges, using respectively their finite $q$th moments and arbitrary Gaussian moments, show both maxima are $o(n^{1\mathrm{/}p})$, as in step 2.1. This proves the stated full-prefix approximation.

In a fixed orthonormal eigenbasis of $\mathrm{\Gamma }$, divide each positive-variance Gaussian coordinate by its fixed standard deviation and add independent standard Gaussian coordinates in the zero-variance directions. This yields independent standard normal vector increments realizing $G_i$ by ${\mathrm{\Gamma }}^{1\mathrm{/}2}$. For each integer interval adjoin independent Brownian copies by [F8] and use their bridges $W(t)-tW(1)$, independently of the existing increments. Interpolate the standard Gaussian random walk linearly on that interval plus its bridge. The paths are continuous and direct covariance calculation gives $\min (s,t)I_d$; finite Gaussianity and zero cross covariances give independent normal increments by [F5]. Thus [F8] identifies a standard $d$-dimensional Brownian motion, and the fixed eigenbasis map gives ${\sum }_{i<k}{G}_i={\mathrm{\Gamma }}^{1\mathrm{/}2}B(k)$ at every integer $k$. This construction also applies when $\mathrm{\Gamma }=0$ by adjoining an independent Brownian motion. It does not alter the original-process marginal or any error already proved.