Tail and moment estimates for a class of random chaoses of order two

Meller, Rafał

doi:10.4064/sm170819-2-5

Cited by 2 publications

(3 citation statements)

References 11 publications

(29 reference statements)

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“…On the other hand, for measures with log-concave tails similar two-sided estimates were proven in [10,20,21,23,3] under different conditions. Moreover, two-sided estimates for non-negative random variables have been derived in [25] and for chaos of order two in symmetric random variables satisfying the inequality X 2p ≤ A X p in [26].…”

Section: Discussion Of Related Literaturementioning

confidence: 99%

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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: Discussion Of Related Literaturementioning

confidence: 99%

“…Especially in the case α = 2 d this yields the existence of the Ψ 1/d norm. However, we want to stress that the results in [3,18,25,26] are two-sided and require very different tools.…”

Section: Discussion Of Related Literaturementioning

confidence: 99%

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

Götze

Sambale

Sinulis

2021

Electron. J. Probab.

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“…More work on related topics include [17,18] where upper and lower bounds for the case of random variables satisfying the moment condition X 2 p ≤ α X p are considered for the case of positive variables of order 2. The recent work [19] provides similar bounds to [6] for distributions of bounded ψ α norm for α ∈ (0, 1] (or α ∈ (0, 2] for some fo their results), such as subexponential distributions.…”

Section: Previous Workmentioning

confidence: 99%

The Hanson–Wright inequality for random tensors

Bamberger

Krahmer

Ward

2022

Sampl. Theory Signal Process. Data Anal.

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We provide moment bounds for expressions of the type $$(X^{(1)} \otimes \cdots \otimes X^{(d)})^T A (X^{(1)} \otimes \cdots \otimes X^{(d)})$$ ( X ( 1 ) ⊗ ⋯ ⊗ X ( d ) ) T A ( X ( 1 ) ⊗ ⋯ ⊗ X ( d ) ) where $$\otimes $$ ⊗ denotes the Kronecker product and $$X^{(1)}, \ldots , X^{(d)}$$ X ( 1 ) , … , X ( d ) are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on d for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form $$\Vert B (X^{(1)} \otimes \cdots \otimes X^{(d)})\Vert _2$$ ‖ B ( X ( 1 ) ⊗ ⋯ ⊗ X ( d ) ) ‖ 2 .

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Tail and moment estimates for a class of random chaoses of order two

Cited by 2 publications

References 11 publications

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

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

The Hanson–Wright inequality for random tensors

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