This work presents a comprehensive oscillation theory for a class of second-order nonlinear neutral differential equations with a variable damping term and variable delay functions. The primary contribution consists of a set of new oscillation criteria whose validity depends on the interplay between the key parameters $\alpha$ and $\beta$. To address the limitations of existing approaches, a novel framework is introduced based on the construction of parameterized auxiliary equations and the use of a generalized Riccati transformation. This approach reduces the problem to the analysis of an associated integro-differential inequality, which is then analyzed via refined inequality techniques and iterative integration. The main results reveal that oscillatory behavior is dictated by a threshold condition comparing $\alpha$ with the rate of change of the delay, quantified by $\beta$. These results are shown to be both sharp and easily applicable, as illustrated by examples that also highlight their improvement over existing results.

In this work, the law of the single logarithm for randomly weighted sums of widely orthant dependent random variables was studied, which improved and generalized some corresponding results in the literature. As an application, the rate of complete convergence for random weighting estimation with widely orthant dependent samples was obtained under general assumptions. Some numerical analyses were also conducted to confirm the theoretical results.