  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 probability 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.
