General Random Sum Limit Theorems for Martingales with large $\mathcal{O}$-Rates

Abstract

This paper is concerned with large $\mathcal{O}$ error estimates for convergence in distribution of random sums of not necessarily independent random variables. As applications of a general theorem one obtains the random-sum versions of the central limit theorem and of the weak law of large numbers for martingale difference sequences by specializing the limiting random variable. Both theorems are equipped with $\mathcal{O}$-rates.