Bounds in Poisson Approximation for Random Sums of Bernoulli Random Variables

Let $(X_n)$ be a sequence of Bernoulli random variables and $N$ a positive integer value random variable. Assume that $N, X_1, X_2,ldots$ are independent. In this paper, we investigate uniform and non-uniform bounds in Poisson approximation for random sums $X_1+X_2+cdots+X_N$.