Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 168 / 4 / i / Solution

Write the multilinear polynomial as $f=g+x_{n}h$, where $g$ is independent of $x_n$, $\mathrm{deg}g\le k$, and $\mathrm{deg}h\le k-1$. Put $m_3=\mathbb{E}{X}_n^3$. The Cauchy-Schwarz inequality gives $|m_3|\le \sqrt{C}$, and independence givesChoose $A=A(C)\ge 1$ so large thatInduction on $n$, using $2a{b}^3\le a^2{b}^2+b^4$ and the orthogonal identity $\Vert f{\Vert }_2^2=\Vert g{\Vert }_2^2+\Vert h{\Vert }_2^2$, now yields

If $\mathbb{E}{X}_i^3=0$ and $C\le 9$, the cubic term vanishes. Taking $A=3$ and applying the induction hypothesis gives exactly

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