A Non-asymptotic, Sharp, and User-friendly Reverse Chernoff-Cramèr Bound

The Chernoff-Cram\`er bound is a widely used technique to analyze the upper tail bound of random variable based on its moment generating function. By elementary proofs, we develop a user-friendly reverse Chernoff-Cram\`er bound that yields non-asymptotic lower tail bounds for generic random variables. The new reverse Chernoff-Cram\`er bound is used to derive a series of results, including the sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which matches the classic Hoefflding-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 develop 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 sparse signal identification is finally studied.