Sharp Concentration Results for Heavy-Tailed Distributions

The concentration of measure inequalities have received substantial attention in high-dimensional statistics and machine learning [1]. While concentration inequalities are well-understood for subGaussian and subexponential random variables, in many application areas, such as signal processing [2] and machine learning [3] we need concentration results for sums of random variables with heavier tails. The standard technique, i.e. finding upper bounds for the moment generating function (MGF), clearly fails for heavy-tailed distributions whose moment generating functions do not exist. Furthermore, other techniques, such as Chebyshev's inequality, are incapable of obtaining sharp results. The goal of this paper is to show that under quite general conditions on the tail a simple truncation argument can not only help us use the standard MGF argument for heavy-tailed random variables, but is also capable of obtaining sharp concentration results.