On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables

The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper tail bounds in literature, the lower tail bound results are relatively fewer. In this partly expository paper, we introduce systematic and user-friendly schemes for developing non-asymptotic lower tail bounds with elementary proofs. In addition, we develop sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which matches 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-squared, 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 studied.