Finite-moment iid vectors admit a strong Gaussian approximation

Put $V=\mathrm{ran}\mathrm{\Sigma }$. For each vector $v$ in a deterministic finite orthonormal basis of $\ker \mathrm{\Sigma }$, $E(v^T{X}_i{)}^2=0$, so $v^T{X}_i=0$ almost surely. Intersect over that basis and the countably many indices to get one full event on which every $X_i\in V$. If $V=\{0\}$, take $G_i=0$ and adjoin an independent standard two-dimensional Brownian motion by [F8]; both approximation errors vanish. Otherwise use a deterministic orthonormal eigenbasis of $\mathrm{\Sigma }$ on $V$ and work in dimension $d=1$ or $2$, preserving Euclidean norms. Its covariance, still denoted $\mathrm{\Sigma }$, is positive definite, with smallest eigenvalue $\lambda >0$ by [F2]. The final construction will be mapped back by the same isometry.

Let $M=E\mathrm{\parallel }{X}_1{\mathrm{\parallel }}^q$, $N_m=2^m$, $I_m=\{2^m,\dots ,2^{m+1}-1\}$, and $L_m=N_m^{1\mathrm{/}a}$. Set ${\mu }_m=E[X_1{\mathbf{1}}_{\{\mathrm{\parallel }{X}_1\mathrm{\parallel }\le L_m\}}]$ and $W_{m,i}=X_i{\mathbf{1}}_{\{\mathrm{\parallel }{X}_i\mathrm{\parallel }\le L_m\}}-{\mu }_m$ for $i\in I_m$. The pointwise bound $x^s{\mathbf{1}}_{\{x>L\}}\le L^{s-q}{x}^q$ for $0\le s\le q$ gives $\mathrm{Pr}(\mathrm{\exists }i\in I_m:\mathrm{\parallel }{X}_i\mathrm{\parallel }>L_m)\le M{N}_m{L}_m^{-q}=M{2}^{-m(q\mathrm{/}a-1)}$. Since $q>a$, this is summable in $m$, and [F3] gives no discarded observations in every sufficiently large block almost surely. Also $E{X}_1=0$ implies $\mathrm{\parallel }{\mu }_m\mathrm{\parallel }\le M{L}_m^{1-q}$. On a block without discarded observations the maximum partial-block difference between $X_i$ and $W_{m,i}$ is at most $M{N}_m{L}_m^{1-q}=M{N}_m^{1+(1-q)\mathrm{/}a}=O(N_m^{1\mathrm{/}a})$, since $a<q$.

The truncated covariance ${\mathrm{\Sigma }}_m=E{W}_{m,i}{W}_{m,i}^T$ satisfies the exact identity $\mathrm{\Sigma }-{\mathrm{\Sigma }}_m=E[X_1{X}_1^T{\mathbf{1}}_{\{\mathrm{\parallel }{X}_1\mathrm{\parallel }>L_m\}}]+{\mu }_m{\mu }_m^T$. Hence $\mathrm{\parallel }\mathrm{\Sigma }-{\mathrm{\Sigma }}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M{L}_m^{2-q}+M^2{L}_m^{2-2q}\le C{L}_m^{2-q}\to 0$. For all sufficiently large deterministic $m$, its eigenvalues are therefore bounded below by $\lambda \mathrm{/}2$ and above by a fixed constant. The standardized truncated vectors $Z_{m,i}={\mathrm{\Sigma }}_m^{-1\mathrm{/}2}{W}_{m,i}$ are centered iid within each block, with covariance $I_d$ and norm bounded by $R_m=C_0{L}_m\ge 1$, after increasing a fixed $C_0$, because $\mathrm{\parallel }{\mu }_m\mathrm{\parallel }\le E\mathrm{\parallel }{X}_1\mathrm{\parallel }$ and $\mathrm{\parallel }{\mathrm{\Sigma }}_m^{-1\mathrm{/}2}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \sqrt{2\mathrm{/}\lambda }$.

Apply [F1] with $A=3$ separately on each such block, obtaining iid standard Gaussian vectors $H_{m,j}$, $1\le j\le N_m$. Put ${\tilde{G}}_{m,j}={\mathrm{\Sigma }}_m^{1\mathrm{/}2}{H}_{m,j}$. The operator norms of ${\mathrm{\Sigma }}_m^{1\mathrm{/}2}$ are uniformly bounded, so the maximum partial-sum discrepancy between the truncated row and this Gaussian row is at most $C{N}_m^{1\mathrm{/}a}(m+1{)}^3$, except with probability at most $C{2}^{-3m}$. Here $C$ is independent of the varying truncated law and $m$, by the uniform constants in [F1].

Set $D_m={\mathrm{\Sigma }}^{1\mathrm{/}2}-{\mathrm{\Sigma }}_m^{1\mathrm{/}2}$. If $A={\mathrm{\Sigma }}^{1\mathrm{/}2}$ and $C_m={\mathrm{\Sigma }}_m^{1\mathrm{/}2}$, then $A{D}_m+D_m{C}_m=\mathrm{\Sigma }-{\mathrm{\Sigma }}_m$. Taking Frobenius inner product with $D_m$ gives $({\lambda }_{\min }(A)+{\lambda }_{\min }(C_m))\mathrm{\parallel }{D}_m{\mathrm{\parallel }}_F^2\le \mathrm{\parallel }{D}_m{\mathrm{\parallel }}_F\mathrm{\parallel }\mathrm{\Sigma }-{\mathrm{\Sigma }}_m{\mathrm{\parallel }}_F$. Treating $D_m=0$ separately and otherwise dividing, the eigenvalue bounds of step 3.1 and equivalence of these matrix norms in dimension at most two give $\mathrm{\parallel }{D}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le C\mathrm{\parallel }\mathrm{\Sigma }-{\mathrm{\Sigma }}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le C{L}_m^{2-q}$. This uses positive definiteness only on $V$, with no inverse on covariance-kernel directions.

To realize these couplings while preserving the original process, use [F4] on each finite coupling law to obtain a kernel giving its Gaussian block conditioned on its standardized truncated block. Evaluate that kernel at the actually given $(Z_{m,i}{)}_{i\in I_m}$. Apply [F5] with initial coordinate the entire original space $(\mathrm{\Omega },\mathcal{F},P)$ and each subsequent coordinate its finite Gaussian block, the kernel depending on $\omega$ through that block alone. This extension keeps the original marginal. Independence of the original disjoint blocks implies, by integrating finite block rectangles, that the unconditional paired-block laws factor into the finite coupling laws; thus all $H_{m,j}$ across all blocks are independent standard Gaussians. Assign independent Gaussian blocks to the finitely many early blocks before invertibility. Their total discrepancy is finite almost surely. On the resulting extension, [F3] and the summable bounds of step 4.1 give the stated partial-block coupling bound for all sufficiently large blocks almost surely.

Put $G_{m,j}={\mathrm{\Sigma }}^{1\mathrm{/}2}{H}_{m,j}$. Each coordinate of ${\sum }_{j=1}^k{D}_m{H}_{m,j}$, $k\le N_m$, is Gaussian with variance at most $N_m\mathrm{\parallel }{D}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2$. By [F6], a union bound over the $d{N}_m$ coordinates and times bounds the probability that its maximum Euclidean norm exceeds $C_1\mathrm{\parallel }{D}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\sqrt{N_m(m+1)}$ by $2d{2}^m\exp [-C_1^2(m+1)\mathrm{/}(2d)]$, summable in $m$ for fixed sufficiently large $C_1$. If $D_m=0$ the discrepancy is zero. Thus [F3] gives a maximum partial-block covariance-replacement error $O(N_m^{1\mathrm{/}2}{L}_m^{2-q}\sqrt{m+1})$ almost surely. Its power $1\mathrm{/}2+(2-q)\mathrm{/}a$ is less than $1\mathrm{/}a$ because $a<q<2(q-1)$, so this error is $O(N_m^{1\mathrm{/}a}(m+1{)}^3)$.

Combine steps 2.1, 5.1 and 6.1 by triangle inequality. On one full event the maximum partial-block error between $X_i$ and $G_{m,j}$ is at most $C(\omega )2^{m\mathrm{/}a}(m+1{)}^3$ for every $m$, after absorbing finitely many early errors into a finite random constant. Concatenate the independent Gaussian blocks into $(G_i)$. For $2^J\le n<2^{J+1}$, the global maximum is at most $C(\omega ){\sum }_{m=0}^J{2}^{m\mathrm{/}a}(m+1{)}^3\le C_{a}C(\omega )2^{J\mathrm{/}a}(J+1{)}^3=O(n^{1\mathrm{/}a}[\log (en){]}^3)$. Because $1\mathrm{/}a<1\mathrm{/}p$, this bound is $o(n^{1\mathrm{/}p})$. Mapping from $V$ back to ${\mathbb{R}}^2$ preserves the error norm and gives covariance equal to the original $\mathrm{\Sigma }$.

To produce the Brownian driver, retain the independent standardized Gaussian increments in the effective eigenbasis of $\mathrm{\Sigma }$. If $d=1$, adjoin independent standard Gaussian increments in its orthogonal complement by [F7] and [F8]. For each coordinate and each integer interval, adjoin an independent Brownian motion $W_{i,l}$ by [F8], independent of all existing variables, and form its bridge ${\beta }_{i,l}(t)=W_{i,l}(t)-t{W}_{i,l}(1)$, $0\le t\le 1$. The bridges are centered Gaussian with covariance $\min (s,t)-st$ and zero endpoints. Interpolate each standardized Gaussian random walk on $[k,k+1]$ by its linear increment plus ${\beta }_{k+1,l}$, independently for the two coordinates. By [F7], finite stacks are Gaussian; direct covariance calculation gives covariance $\min (s,t)$ within each coordinate and zero cross-coordinate covariance. Paths are continuous, including at integer joins. The covariance calculation makes vector increments over disjoint time intervals uncorrelated; finite joint Gaussianity and [F9] make them independent with covariance equal to interval length times $I_2$, so [F9] identifies standard two-dimensional Brownian motion. Applying the deterministic orthogonal eigenbasis map makes a standard two-dimensional $B$ and yields ${\sum }_{i=1}^k{G}_i={\mathrm{\Sigma }}^{1\mathrm{/}2}B(k)$ for every integer $k$. Step 7.1 supplies the asserted Brownian approximation and its weaker $O(n^{1\mathrm{/}p}\log n)$ rate.