A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables

Hanson, D. L.; Wright, F. T.

doi:10.1214/aoms/1177693335

Cited by 312 publications

(291 citation statements)

References 2 publications

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“…It is therefore important to obtain two-sided bounds in terms of purely deterministic quantities. Such bounds for random quadratic forms in independent symmetric random variables with log-concave tails have been obtained by Latała [43] (the case of linear forms was solved earlier by Gluskin and Kwapień [29], whereas bounds for quadratic forms in Gaussian variables were obtained by Hanson-Wright [32], Borell [15] and Arcones-Giné [5]). Their counterparts for multilinear forms of arbitrary degree in nonnegative random variables with log-concave tails have been derived by Latała and Łochowski [45].…”

Section: P(| F (G) − E F (G)| ≥ T) ≤ 2 Exp(−t 2 /2)mentioning

confidence: 86%

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

Adamczak

Wolff

2014

Probab. Theory Relat. Fields

163

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Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalitiesSuch Sobolev type inequalities hold, e.g., if the underlying measure satisfies the logSobolev inequality (in which case C( p) ≤ C √ p) or the Poincaré inequality (then C( p) ≤ C p). 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 (not 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-Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.

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Section: P(| F (G) − E F (G)| ≥ T) ≤ 2 Exp(−t 2 /2)mentioning

confidence: 86%

“…To give the Reader a flavour of possible applications let us mention the HansonWright inequality [32]. Namely, for a random vector X = (X 1 , .…”

mentioning

confidence: 99%

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

Adamczak

Wolff

2014

Probab. Theory Relat. Fields

163

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“…These types of inequalities are already present in Hoeffding seminal papers [6], [7] and have seen further development since then. For example, exponential bounds were obtained (in the (sub)Gaussian case) by Hanson and Wright [5], by Bretagnolle [1], and most recently by Giné, Lata la, and Zinn [4] (and the many references therein). As indicated in [4], the exponential bound there is optimal since it involves a mixture of exponents corresponding to a Gaussian chaos of order two behavior, and (up to logarithmic factors) to the product of a normal and of a Poisson random variable and to the product of two independent Poisson random variables.…”

Section: Introductionmentioning

confidence: 99%

Exponential Inequalities, with Constants, for U-statistics of Order Two

Houdré

Reynaud-Bouret²

2003

Stochastic Inequalities and Applications

103

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“…It also implies n exp( [14] obtained the first important inequality for sub-gaussian random variables.…”

Section: Proofs Of New Results Let Us First Prove Theorem 14 Noticmentioning

confidence: 90%

Random weighted projections, random quadratic forms and random eigenvectors

Wang

2014

Random Struct. Alg.

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A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables

Cited by 312 publications

References 2 publications

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

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

Exponential Inequalities, with Constants, for U-statistics of Order Two

Random weighted projections, random quadratic forms and random eigenvectors

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