Heterogeneous bounded atom vector rows admit a logarithmic cubed Gaussian coupling

For a coordinate $Y$ of one centered atom, $E\mathrm{\mid }Y{\mathrm{\mid }}^r\le B^{r-2}E{Y}^2$ for every integer $r\ge 2$. Absolute exponential-series integration is legitimate because ${\sum }_r\mathrm{\mid }hY{\mathrm{\mid }}^r\mathrm{/}r!\le e^{B\mathrm{\mid }h\mathrm{\mid }}$; [F6] permits summing its nonnegative absolute terms, and its positive and negative parts then give the signed series identity. For $B\mathrm{\mid }h\mathrm{\mid }<1$, centering cancels the linear term and $r!\ge 2$ for $r\ge 2$, so $\log E{e}^{hY}\le E{e}^{hY}-1\le \frac{h^{2}E{Y}^2}{2(1-B\mathrm{\mid }h\mathrm{\mid })}\mathrm{.}$ A centered Gaussian coordinate has logarithmic mgf $h^2\mathrm{var}(Z)\mathrm{/}2$, by completing the square in its positive-variance eigencoordinates using [F3] and [F4]; zero-variance coordinates vanish. Independence multiplies the finitely many atom and Gaussian mgfs. Since their variances add to each coordinate variance of $X_i$, at most $K$, any sum of $s$ entries, also with arbitrary matching Gaussian replacements, has coordinate log mgf at most $Ks{h}^2\mathrm{/}[2(1-B\mathrm{\mid }h\mathrm{\mid })]$. For a coordinate threshold $u>0$ use $h=u\mathrm{/}(Ks+Bu)$ and exponential Markov for both signs. This gives tail at most $2\exp \{-u^2\mathrm{/}[2(Ks+Bu)]\}$. The coordinate union bound therefore gives, uniformly in atom counts, $\mathrm{Pr}\text{\hspace{0.17em}⁣}(\mid {\sum }_{i\in I}{X}_i\mid >C_K[\sqrt{st}+Bt])\le C_K{e}^{-c_{K}t},t\ge 1.$ This holds for every contiguous interval in any compressed ordering.

Consider a dyadic mixed row whose odd entries are independent matching Gaussians and whose even entries have their raw laws. At a node of $s\ge 4$ entries let its two child covariance matrices be $U,V$, put $W=U+V$, and, for its target child sums $S_L,S_R$, define $P=S_L+S_R,H=U{W}^{-1},D=S_L-HP,R=U-U{W}^{-1}U\mathrm{.}$ Independence gives $\mathrm{cov}(P)=W$, $\mathrm{cov}(P,D)=U-W{W}^{-1}U=0$, and $\mathrm{cov}(D)=R$. The symmetric matrix $R$ equals $U{W}^{-1}V$ and has inverse $V^{-1}W{U}^{-1}=U^{-1}+V^{-1}$; hence it is positive definite without assuming commutation. Since each child has $s\mathrm{/}2$ entries, [F3] gives $\frac{s}{2}I\le U,V\le \frac{Ks}{2}I,sI\le W\le KsI,\frac{s}{4}I\le R\le \frac{Ks}{4}I\mathrm{.}$ The final inequalities follow by inverting, adding, and reinverting the quadratic form bounds.

The transformed pair $Q=\sqrt{s}\mathrm{diag}(W^{-1\mathrm{/}2},R^{-1\mathrm{/}2})(P,D)$ has covariance $s{I}_{2d}$. A raw atom $x$ in its left or right child contributes respectively $\sqrt{s}(W^{-1\mathrm{/}2}x,R^{-1\mathrm{/}2}V{W}^{-1}x),\sqrt{s}(W^{-1\mathrm{/}2}x,-R^{-1\mathrm{/}2}U{W}^{-1}x)\mathrm{.}$ Their norms are at most $C_{K}B$, since $\sqrt{s}\mathrm{\parallel }{W}^{-1\mathrm{/}2}\mathrm{\parallel }\le 1$, $\sqrt{s}\mathrm{\parallel }{R}^{-1\mathrm{/}2}\mathrm{\parallel }\le 2$, and $\mathrm{\parallel }U{W}^{-1}\mathrm{\parallel },\mathrm{\parallel }V{W}^{-1}\mathrm{\parallel }\le K\mathrm{/}2$. Each child contains half Gaussian entries, so its Gaussian covariance is at least $(s\mathrm{/}4)I$, at least $\gamma$ times its total covariance with $\gamma =1\mathrm{/}(2K)$. Congruence of quadratic forms therefore makes the Gaussian covariance of $Q$ at least $\gamma s{I}_{2d}$. All its bounded contributions are the independently transformed original atoms, and all Gaussian contributions combine into an independent Gaussian. Thus [F1] and [F2] apply directly to $Q$, including its coordinate prefixes. The conditioning is on its first $d$ coordinates. Undoing the separate parent and residual whitening, whose inverse norms are bounded by $C_K$, gives a transport of a fresh $E_I\sim N_d(0,R)$ to $D$ conditional on $P=p$, with error $\mathrm{\mid }{T}_{I,p}(E_I)-E_I\mathrm{\mid }\le C_{K}B(1+\frac{\mathrm{\mid }p{\mathrm{\mid }}^2+\mathrm{\mid }{T}_{I,p}(E_I){\mathrm{\mid }}^2}{s})$ when $s\ge C_K{B}^2$ and the two target norms are at most $c_{K}s\mathrm{/}B$. The root has the analogous transport from $N_d(0,W)$ with error $C_{K}B(1+\mathrm{\mid }P{\mathrm{\mid }}^2\mathrm{/}s)$. There is no additional factor from the atom counts or from $\sqrt{s}$.

Temporarily take a dyadic current length $q\le 2N$, set $L=a\log (eN)$, and let $q_0$ be the least dyadic integer at least $H_K{B}^{2}L$, with $H_K$ to be fixed. Suppose $q\ge 2q_0$. Start with an independent Gaussian row having the individual covariances $D_i$. Its root sum and residuals $E_I=V_L-U{W}^{-1}{V}_I$ at the active nodes are mutually independent Gaussian vectors of covariances $W_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$ and $R_I$: the cross covariances vanish, and independence follows recursively from [F4]. Apply the root transport in step 2.1 and then, descending the tree, set $D_I=T_{I,S_I}(E_I)$ and $S_L=U{W}^{-1}{S}_I+D_I,S_R=V{W}^{-1}{S}_I-D_I,V_L=U{W}^{-1}{V}_I+E_I,V_R=V{W}^{-1}{V}_I-E_I\mathrm{.}$ Stop at length $q_0$. Each fresh residual is independent of the already exposed parents, so every target split has exactly its prescribed conditional law. For Gaussians the child covariance is $U{W}^{-1}U+R=U$ and the cross covariance is $U{W}^{-1}V-R=0$, with the analogous right covariance $V$.

There is an explicit terminal target kernel, also measurable in any supplied environment. In a terminal mixed block let $\mu$ be the product law of its raw positions, let $s(r)$ be their sum, and let $G>0$ be the sum of the matching Gaussian-position covariances. Given the target block sum $u$, sample the raw-position vector from $\frac{q_G(u-s(r))\mu (dr)}{\int q_G(u-s(v))\mu (dv)}\mathrm{.}$ The denominator is finite and strictly positive everywhere by [F4]. Fill the Gaussian positions conditional on their remaining sum $u-s(r)$: for their covariance matrices $C_j$ and fresh independent $h_j\sim N(0,C_j)$, put $H_0={\sum }_j{h}_j$ and $g_j=C_j{G}^{-1}(u-s(r))+h_j-C_j{G}^{-1}{H}_0$. The residual vector has sum zero, zero cross covariance with $H_0$, and covariance blocks ${\delta }_{ij}{C}_i-C_i{G}^{-1}{C}_j$; thus [F4] identifies its Gaussian conditional law. Multiplying the posterior by the total-sum density and integrating in $u$ proves the original raw/Gaussian product law, first on rectangles and then by monotone approximation. This verifies the terminal conditional law at every sum, without choosing regular conditional versions. Independent terminal kernels and the split laws of step 3.1 consequently recover exactly the entire mixed row product law by induction through the disjoint binary blocks. The existing Gaussian row is unchanged, and already has its required product law.

Use step 1.1 and a union bound for every dyadic target sum and every within-terminal prefix in the completed target and Gaussian rows. There are at most $Cq$ such vectors. Except on probability $C_{K}q{e}^{-c_{K}L}$, each target sum of size $s$ has norm at most $C_K(\sqrt{sL}+BL)$. The target residual is $D_I=V{W}^{-1}{S}_L-U{W}^{-1}{S}_R$, so its norm satisfies the same bound with a different $C_K$. At active sizes $s\ge 2q_0$, choose $H_K$ large enough that these target norms are at most $c_{K}s\mathrm{/}B$; their ratios to $s\mathrm{/}B$ are bounded by $C_K\mathrm{/}\sqrt{H_K}+C_K\mathrm{/}{H}_K$. Their squared norms divided by $s$ are at most $C_{K}L$. Step 2.1 therefore bounds the root discrepancy and every residual discrepancy ${\delta }_I=D_I-E_I$ by $C_{K}BL$. The within-terminal prefix norms are at most $C_K(\sqrt{q_{0}L}+BL)\le C_{K}BL$, since $q_0<2H_K{B}^{2}L$. No conditional concentration inside a terminal block has been assumed.

Write $e_I=S_I-V_I$. Its child equations are $e_L=U{W}^{-1}{e}_I+{\delta }_I$ and $e_R=V{W}^{-1}{e}_I-{\delta }_I$. For a prefix $P$ ending at a terminal boundary of $I$, let $C_P$ be the sum of its individual covariance matrices. Its discrepancy has the exact expansion ${\sum }_{i\in P}(X_i-G_i)=C_P{W}_I^{-1}{e}_I+{\sum }_{J\text{ on the boundary path}}{M}_{J,P}{\delta }_J\mathrm{.}$ If the boundary lies in the left child, $M_{J,P}=C_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}{W}_{J_L}^{-1}$; if it lies in the right child, $M_{J,P}=I-C_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{W}_{J_R}^{-1}$. To prove the formula, in the left case substitute the child's leading term $C_P{U}^{-1}{e}_L$ and cancel $U^{-1}U$; in the right case add $e_L$ to $C_Q{V}^{-1}{e}_R$, obtaining leading coefficient $(U+C_Q)W^{-1}$ and residual coefficient $I-C_Q{V}^{-1}$. Induction from terminal blocks proves the expansion. This cancellation uses the order $W_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{c}}{W}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}^{-1}{W}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}^{-1}$ and never commutes two matrices. If a partial child has $r$ of its $h$ entries, its coefficient norm is at most $Kr\mathrm{/}h\le K$. Hence $\mathrm{\parallel }{M}_{J,P}\mathrm{\parallel }\le 1+K$ and $\mathrm{\parallel }{C}_P{W}_I^{-1}\mathrm{\parallel }\le K$. There are at most ${\log }_{2}q$ boundary ancestors. The expansion and step 5.1 give terminal-boundary error $C_{K}BL\log (eq)$; adding its terminal prefix residual gives the same bound for all prefixes.

If $q<2q_0$, couple a raw or mixed row independently to a matching Gaussian row. Their unconditional concentration bounds from step 1.1 and a union bound over prefixes give maximum discrepancy at most $C_K(\sqrt{qL}+BL)\le C_{K}BL$ except on probability $C_{K}q{e}^{-c_{K}L}$. This covers the small sizes, including a row of one entry, and large $B$. Thus in all cases a mixed row can be coupled directly to an already realized matching Gaussian row with stage error at most $C_{K}BL\log (eq)$ and bad probability at most $C_{K}q{e}^{-c_{K}L}$.

Pad the initial length $m$ to dyadic $q<2m$ by independent $N_d(0,I_d)$ entries; for $m=1$ take $q=1$. These added laws have empty atom sums and preserve all covariance bounds. Let $R^{(0)}$ be the matching independent Gaussian row, saved as the final reference row, and initially mark every position active. Inductively, the current full row consists of independent frozen raw entries and an independent active Gaussian row, all with the designated covariances. When its active length is at least $2q_0$, use that active Gaussian row and fresh terminal randomness in steps 3.1--6.1 to construct the target mixed row, with designated raw laws in even slots and matching Gaussian laws in odd slots. Replace the entire active row by this target row, including its odd coordinates, to obtain $R^{(j+1)}$ from $R^{(j)}$. Freeze the target evens and use the target Gaussian odds as the next active row. The extension kernel depends only on the previous active row, so [F7] makes the entire target independent of the previously frozen complement; its product marginal from step 4.1 makes its new raw evens independent of its Gaussian odds. This proves the induction. At active length below $2q_0$, replace every remaining active coordinate by independent designated raw entries using step 7.1 and stop. After at most $C\log (eN)$ stages, the final row $R^{(J)}$ has the intended raw product law. The saved row $R^{(0)}$ retains its Gaussian product law, although subsequent active Gaussian rows need not equal its corresponding coordinates.

Write ${\mathrm{\Delta }}_j(k)={\sum }_{i\le k}(R_i^{(j+1)}-R_i^{(j)})$. Frozen entries do not change at stage $j$, and every original prefix meets its active positions in an initial compressed prefix. Thus the stage estimate in steps 6.1--7.1 bounds ${\max }_k\mathrm{\mid }{\mathrm{\Delta }}_j(k)\mathrm{\mid }$ by $C_{K}BL\log (eN)$ outside its stage bad event; this includes all changes to target Gaussian odds. The exact identity ${\sum }_{i\le k}(R_i^{(J)}-R_i^{(0)})={\sum }_{j=0}^{J-1}{\mathrm{\Delta }}_j(k)$ and triangle inequality therefore give total error at most $C_{A,K}B{\log }^3(eN)$. Each stage has the Gaussian input marginal and mixed target marginal used in its estimate by step 8.1, so its unconditional bad probability remains at most $C_K{q}_j{e}^{-c_{K}L}$, where $q_j\le 2N$ is its active length. With the same ambient $L=a\log (eN)$, a union bound over at most $C\log (eN)$ stages gives $C_{K}N\log (eN)e^{-c_{K}a\log (eN)}$. Choose $a$ sufficiently large depending only on $A,K$ to make this at most $N^{-A}$ for all $N\ge 1$. Taking $X=R^{(J)}$, $G=R^{(0)}$ and restricting to the first $m$ entries proves the numerical assertion.

For the parameter assertion, every entry law is a measurable kernel obtained by applying addition to the finite product of its supplied atom and Gaussian kernels, using [F5]. Covariance entries are measurable integrals. In dimensions one and two positive definite inverse square roots are continuous functions of their matrices: in dimension two, for $C>0$, $C^{1\mathrm{/}2}=(C+\sqrt{\det C}I)\mathrm{/}\sqrt{\mathrm{tr}C+2\sqrt{\det C}}$, as verified in an eigenbasis by [F3]; the semidefinite zero case is defined by zero and the formula extends continuously. Thus the Gaussian starts and all transforms in steps 1.2, 2.1 depend measurably on the parameter. Mixed transformed joint and prefix densities are Gaussian convolution integrals of their raw product kernels, so are jointly measurable by [F5] and strictly positive. Their conditional CDFs are measurable density-ratio integrals. The coordinate quantiles in [F2] are measurable rational infima of those CDFs; hence its root and residual maps are jointly measurable in the parameter and all spatial inputs. The terminal posterior in step 4.1 is likewise a measurable kernel with positive measurable denominator, followed by the explicit Gaussian bridge kernel. Finite kernel products and successive extensions preserve measurability: for one extension define its law by integrating the next kernel against the preceding law; [F5] gives measurable evaluations and [F6] gives countable additivity by monotone convergence of disjoint indicator sums. Iterating this finite construction gives a coupling kernel $Q_{\theta }$, and every previous marginal and probability bound holds at each parameter. No pointwise selection of conditional-law versions or coupling laws is used.