In this paper we propose a novel approach for Twitter traffic analysis based on renewal theory. Even though twitter datasets are of increasing interest to researchers, extracting information from message timing remains somewhat unexplored. Our approach, extending our prior work on anomaly detection, makes it possible to characterize levels of correlation within a message stream, thus assessing how much interaction there is between those posting messages. Moreover, our method enables us to detect the presence of periodic traffic, which is useful to determine whether there is spam in the message stream. Because our proposed techniques only make use of timing information and are amenable to downsampling, they can be used as low complexity tools for data analysis.