Worst-case Chernoff bounds for sums of random variables with known means and unequal ranges

We construct a new tail bound for the sum of independent random variables for situations in which the expected value of the sum is known and each random variable lies within a specified interval, which may be different for each variable. This new bound can be computed by solving a two-dimensional convex optimization problem. Simulations demonstrate that the new bound is often substantially tighter than Hoeffding’s inequality for cases in which both bounds are applicable.