Invariance principle for a low-degree multilinear polynomial
= 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 $\lVert\psi^{(4)}\rVert_\infty\leq M$, then
$$
|\mathbb E\psi(f(X))-\mathbb E\psi(f(Y))|
\leq\frac{M}{12}9^k\sum_i\operatorname{Inf}_i(f)^2.
$$