This is a consideration paper on the applying of the stream theory for the era of big data analytics. The stream theory that includes the stream algebra and the stream coalgebra, and this theory plays a very important role in mathematics and computer science, particularly in the analytics of big data in generally and big data in livestream (BDL) in particularly. Both the stream algebra and the stream coalgebra are applied to solving problems involving structured, semi-structured, and unstructured data. This research focuses on analyzing and comparing the stream theory based on characteristic criteria. This research constructs eight tables to analyze and compare the characteristic criteria that the stream theory has contributed from the past to the present. Through these analytical and comparative tables, we can gain an extensive understanding of the contributions of the stream theory and help them envision the development of the stream theory in the future.