We consider discrete random variables \(X\) whose probability generating functions are zero-free in a certain sector around the positive real axis in the complex plane. We give sharp bounds on the cumulants of all orders of \(X\) . By the method of cumulants, these bounds lead to a Berry-Essen type CLT, mod-Gaussian convergence, moderate deviation results and concentration inequalities. We also provide an alternative proof for a class of random variables where the zeros of the probability generating functions are located on the unit circle. Compared to the existing results, this leads to an improved bound. A few examples will be discussed.