Generalized order statistics provide a unified framework for various models of ordered random variables, encompassing ordinary order statistics, nonintegral sample size order statistics, progressively type II censored order statistics, record values, and kth record values, assuming the underlying distribution function is continuous. Much research has focused on the continuous case. This work establishes the distribution theory of generalized order statistics for any arbitrary distribution function and offers characterizations of discrete distributions through properties such as independence, conditional independence, and conditional expectations of these statistics. The concept of generalized order statistics illustrates different models of ordered random variables within a broader framework, including ordinary order statistics and progressively type II censored order statistics, as well as records and k-records concerning their distributions. Previous studies primarily assumed the underlying distribution function to be continuous. This research advances the distribution theory of generalized order statistics based on arbitrary distributions, providing insights into the characterization of discrete probability distributions through their distributional properties and independence relations.
