On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables
This work addresses a gap in probability theory and statistics by providing lower tail bounds, which is important for researchers and practitioners in fields like machine learning and signal processing, though it is incremental as it builds on existing upper bound literature.
The paper tackles the problem of developing non-asymptotic lower tail bounds for random variables, which are less studied than upper bounds, and introduces systematic schemes for this, achieving sharp bounds that match classic concentration inequalities like Hoeffding-type and Bernstein-type for sums of independent sub-Gaussian and sub-exponential variables.
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.