 Solution

The relevant invariance principle for a low-degree multilinear polynomial is the following. Let $(X_i)$ and $(Y_i)$ be sequences of independent random variables satisfyingIf $f$ is multilinear of degree at most $k$ and $\Vert {\psi }^{(4)}{\Vert }_{\infty }\le M$, then

For the proof, use the Lindeberg replacement method. Replace $X_i$ by $Y_i$ one coordinate at a time and write $f=u_i+z_i{v}_i$, where $u_i=E_{i}f$ and $v_i=D_{i}f$ depend only on the other coordinates. A third-order Taylor expansion of $\psi (u_i+z_i{v}_i)$ has identical expected terms through order three for $z_i=X_i$ and $z_i=Y_i$, because the first three moments match. Each fourth-order remainder is bounded by $M|z_i{v}_i{|}^4/24$, so the $i$th replacement costs at mostPart (i), applied in the hybrid product space, bounds each of the two fourth-moment terms by $9^k{\mathrm{Inf}}_i(f{)}^2$. Thus the cost is at most $(M/12)9^k{\mathrm{Inf}}_i(f{)}^2$. The triangle inequality and summation over $i$ prove the result.

Solved by gpt-5.6-sol high.