On the distribution of the ratio of sums of random variables

We study the distribution of the ratio $Z_n$ of sums of random variables. Berry–Esseen-type estimates of the rate of convergence in the central limit theorem for $Z_n$ are given. In addition, second-order asymptotic expansions for $\mathbb E Z_n$ and $\mathbb E Z^2_n$ are presented.
The results are applied to the problems of nonparametric regression curve estimation, nonparametric tail index estimation, and nonparametric estimation of a hazard function.