This paper addresses the approximation of partial sums of independent random variables (RVs) by sums of independent standard normal variables. The authors introduce a new construction for the pair of sums, proving that if the RVs have a finite moment generating function and satisfy certain conditions, the approximation error is bounded by \(O(\log n)\) with probability one. This result improves upon previous bounds and provides a method for approximating sample distribution functions. The construction involves a diadic approximation, where the RVs are divided into blocks and approximated using quantile transformations. The proof of the main theorem relies on large deviation principles and the central limit theorem, showing that the error in the approximation decreases as \(O(\log n)\). The paper also discusses extensions to the case of empirical distribution functions and Brownian bridges, providing bounds on the approximation errors in these contexts.This paper addresses the approximation of partial sums of independent random variables (RVs) by sums of independent standard normal variables. The authors introduce a new construction for the pair of sums, proving that if the RVs have a finite moment generating function and satisfy certain conditions, the approximation error is bounded by \(O(\log n)\) with probability one. This result improves upon previous bounds and provides a method for approximating sample distribution functions. The construction involves a diadic approximation, where the RVs are divided into blocks and approximated using quantile transformations. The proof of the main theorem relies on large deviation principles and the central limit theorem, showing that the error in the approximation decreases as \(O(\log n)\). The paper also discusses extensions to the case of empirical distribution functions and Brownian bridges, providing bounds on the approximation errors in these contexts.