In this talk we look at the problem of sketching distributions in the data-stream model. This is a model that has become increasingly popular over the last ten years as practitioners in a variety of areas have sought to design systems that handle massive amounts of data in a time and space efficient manner. Problems such as estimating the distance between two streams, testing independence or identifying correlations, and determining if a distribution is compressible play an important role.

We start by reviewing results on using $p$-stable distributions to compute small-space sketches that can be used to estimate the $L_p$ distance between two distributions. We then present recent results on extending this work to estimate the strength of correlations between two distributions. We finish with an overview of work that seeks to characterize the limits of these techniques with a particular emphasis on what is possible in regards to sketching information divergences such as the Kullback-Leibler, Jensen-Shannon, and Hellinger divergences.