Sharp Tail Bounds Beyond Twice the Mean

Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: $$\mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,.$$ To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\mathbb{P}\left[X_i=0\right]= 1-\frac{1}{t}$ and $\mathbb{P}\left[X_i=t\right]=\frac{1}{t}$, which remains the maximizer in the relaxed problem.