Deviation bounds for the norm of a random vector under exponential moment conditions with applications  

作　　者：Vladimir Spokoiny 

机构地区：[1]Weierstrass Institute,HU Berlin,and HSE Moscow,Mohrenstr.39,10117 Berlin,Germany

出　　处：《Probability, Uncertainty and Quantitative Risk》2025年第1期135-158,共24页概率、不确定性与定量风险(英文)

基　　金：Financial support by the German Research Foundation(DFG)through the Collaborative Research Center 1294"Data assimnilation"is gratefully acknowledged.

摘　　要：Hanson-Wright inequality provides a powerful tool for bounding the norm ||ζ|| of a centered stochastic vector ζ with independent entries and sub-gaussian behavior.This paper extends the bounds to the case when ζ only has bounded exponential moments of the form log E exp<V^(-1)ζ,u>≤||u||^(2)/2,where V^(2)≥Var(ζ)and ||u||≤g for some fixed g.For a linear mapping Q,we present an upper quantile function z_(c)(B,x)ensuring P(||Qζ||>z_(c)(B,x))≤3e^(-x) with B=QV^(2)Q^(T).The obtained results exhibit a phase transition effect:with a value Cc depending on g and B,for x≤x_(c),the function z_(c)(B,x)replicates the case of a Gaussian vector ζ,that is,z_(c)^(2)(B,x)=tr(B)+2√xtr(B_(2))+2x||B||.For x>x_(c),the function z_(c)(B,x)grows linearly in.The results are specified to the case of Bernoulli vector sums and to covariance estimation in Frobenius norm.

关 键 词：Upper quantiles Phase transition Vector Bernoulli sums Frobenius loss 

分 类 号：O211[理学—概率论与数理统计]