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On the non‐asymptotic and sharp lower tail bounds of random variables
Author(s) -
Zhang Anru R.,
Zhou Yuchen
Publication year - 2020
Publication title -
stat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.61
H-Index - 18
ISSN - 2049-1573
DOI - 10.1002/sta4.314
Subject(s) - mathematics , random variable , upper and lower bounds , poisson distribution , gaussian , combinatorics , binomial (polynomial) , exponential family , matching (statistics) , sum of normally distributed random variables , exponential function , type (biology) , discrete mathematics , statistics , multivariate random variable , mathematical analysis , ecology , physics , quantum mechanics , biology
The non‐asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probabilities in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user‐friendly schemes for developing non‐asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub‐Gaussian and sub‐exponential random variables, which match the classic Hoeffding‐type and Bernstein‐type concentration inequalities, respectively. We also provide non‐asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi‐square, binomial, Poisson, Irwin–Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub‐Gaussian and sub‐exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally considered.