Bernstein-like Concentration and Moment Inequalities for Polynomials of Independent Random Variables: Multilinear Case

We show that the probability that a multilinear polynomial $f$ of independent random variables exceeds its mean by $λ$ is at most $e^{-λ^2 / (R^q Var(f))}$ for sufficiently small $λ$, where $R$ is an absolute constant. This matches (up to constants in the exponent) what one would expect from the central limit theorem. Our methods handle a variety of types of random variables including Gaussian, Boolean, exponential, and Poisson. Previous work by Kim-Vu and Schudy-Sviridenko gave bounds of the same form that involved less natural parameters in place of the variance.

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