On the non-asymptotic and sharp lower tail bounds of random variables

Anru R. Zhang, Yuchen Zhou

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Article numbere314
JournalStat
Volume9
Issue number1
DOIs
StatePublished - 2020
Externally publishedYes

Keywords

  • Chernoff–Cramèr bound
  • concentration inequality
  • sub-exponential distribution
  • sub-Gaussian distribution
  • tail bound

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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