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Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nabla g\|_p.$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincaré inequality (then $C(p) \le Cp$). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of $f$. When the underlying measure is Gaussian and $f$ is a polynomial (non-necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-R{é}nyi random graphs, obtaining new estimates, optimal in a certain range of parameters.