Let $X_1,\cdots,X_n$ be independent centered univariate random variables. Let also $\{w_{ij}\}_{i,j=1}^{k,n}$ be a set of deterministic scalar weights, where $k\ll n$. Define sample averages $$ \overline Y_i := \sum_{j=1}^n w_{ij}X_j, \;\;\;\;\;i=1,2,\cdots,k. $$ It should be noted that $\mathbb E\overline Y_i=0$, but $\{\overline Y_i\}$ are in general not independent.
Further assume:
- $\mathrm{cov}(\overline Y)$ is non-singular and finite;
- $\sum_{j=1}^n\mathbb E\|w_{ij}X_j\|_2^3 \leq M <\infty$ for all $i=1,2,\cdots,k$.
My question is the following:
Let $Z$ be a $k$-dimensional centered normal random vector such that $\mathrm{cov}(Z)=\mathrm{cov}(\overline Y)$. Does there exist a constant $C(k,M)$ depending only on $k$ and $M$ such that, for all symmetric $Q\in\mathbb R^{k\times k}$ and vector $a\in\mathbb R^k$, $$ \sup_{t\in\mathbb R} \big|\Pr[\psi_{Q,a}(\overline Y)\leq t] - \Pr[\psi_{Q,a}(Z)\leq t]\big| \leq C(k,M)/\sqrt{n}? $$ Here $\psi_{Q,a}(Z):=\frac{1}{2}Z^\top QZ + a^\top Z$ is a quadratic function of $Z$. Note that the constant $C(k,M)$ must be independent of $Q$ and $a$. Furthermore, it is desired that $C(k,M)$ is a polynomial function of $k$.
Some remarks on special cases where the above invariance principle is true.
- If $Q$ is positive semi-definite, then the image of $\psi_{Q,a}$ for any $t$ is a convex set in $\mathbb R^k$. The claim then holds by invoking the results of Gotze 1991, which upper bounds $|\Pr[\overline Y\in A]-\Pr[\overline Z\in A]|$ uniformly over all measurable convex $A\in\mathbb R^k$. Furthermore, the constant scales as $C(k,M)=O(kM)$.
- If $k=2$ and $Q$ is zero on its diagonal, one can show that $\psi^{-1}(t)$ is either a union of two convex sets in $\mathbb R^2$ or their complement. Therefore, by invoking again the above-mentioned result together with union bound we can obtain similar claims.
- What I'm mostly interested in is $k$ being even and $Q$ has the structure of $Q=[0\;\; C; C^\top\;\; 0]$, meaning that $Q$ has zeros on the diagonal. For $k>2$, it is unclear the image of $\psi$ can be written as a small number of convex sets.
- Finally, it is tempting to use the invariance principle for low-degree polynomial functions of independent samples (see, e.g., http://annals.math.princeton.edu/2010/171-1/p05), as $\psi_{Q,a}(\overline Y)$ can clearly be written as a quadratic function of $X_1,\cdots,X_n$. However, since it is not assumed that $\mathbb E|X_i|^3$ is small, such appoach is unlikely to succeed.
