Abstract: We study random variables of the form $f(X)$, when $f$ is a degree $d$ polynomial, and $X$ is a random vector on $\mathbb{R}^{n}$, motivated towards a deeper understanding of the covariance structure of $X^{\otimes d}$. For applications, the main interest is to bound $\mathrm{Var}(f(X))$ from below, assuming a suitable normalization on the coefficients of $f$. Our first result applies when $X$ has independent coordinates, and we establish dimension-free bounds. We also show that the assumption of independence can be relaxed and that our bounds ...
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Topics: 
Combinatorics
Pure mathematics
Mathematical analysis