Higher Order Concentration in Presence of Poincaré-Type Inequalities

Götze, Friedrich; Sambale, Holger

doi:10.1007/978-3-030-26391-1_6

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2021

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“…First, up to constants, Corollary 1.4 (1) gives back a classical result for the tails of a linear form in random variables with sub-exponential tails for α = 1. For more general functions and similar results under a Poincaré-type inequality, we refer to [7] (the first order case) and [11] (the higher order case).…”

Section: Resultsmentioning

confidence: 99%

Concentration inequalities for polynomials in α-sub-exponential random variables

Götze

Sambale

Sinulis

2021

Electron. J. Probab.

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We derive multi-level concentration inequalities for polynomials in independent random variables with an α-sub-exponential tail decay. A particularly interesting case is given by quadratic forms f (X1, . . . , Xn) = X, AX , for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix A. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in α-sub-exponential random variables, such as quadratic Poisson chaos.We provide various applications of these inequalities. Among them are generalizations of some results proven by Rudelson and Vershynin from sub-Gaussian to α-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector and concentration inequalities for the distance between a random vector and a fixed subspace.

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Section: Resultsmentioning

confidence: 99%