Invariance principle for a low-degree multilinear polynomial

Let $X_i$ and $Y_i$ be independent, centered, variance-one random variables with vanishing third moments and fourth moments at most $9$. If $f$ is multilinear of degree at most $k$ and $\Vert {\psi }^{(4)}{\Vert }_{\infty }\le M$, then

The Lindeberg replacement method compares functions of two independent random vectors by replacing their coordinates one at a time. Matching moments cancel the corresponding terms in a Taylor expansion.