A fully polynomial-time approximation scheme for approximating a sum of random variables

Given n independent integer-valued random variables X1, X2, ..., Xn and an integer C, we study the fundamental problem of computing the probability that the sum X = X1+X2+...+Xn is at most C. We assume that each random variable Xi is implicitly given by an oracle Oi, which given two input integers n1, n2 returns the probability of n1 ≤ Xi ≤ n2. We give the first deterministic fully polynomial-time approximation scheme (FPTAS) to estimate the probability up to a relative error of 1±ǫ. Our algorithm is based on the technique for approximately counting knapsack solutions, developed in [Gopalan et al. FOCS11].