We provide statistical learning guarantees for two unsupervised learning tasks in the context of compressive statistical learning, a general framework for resource-efficient large-scale learning that we introduced in a companion paper. The principle of compressive statistical learning is to compress a training collection, in one pass, into a low-dimensional sketch (a vector of random empirical generalized moments) that captures the information relevant to the considered learning task. We explicitly describe and analyze random feature functions which empirical averages preserve the needed information for compressive clustering and compressive Gaussian mixture modeling with fixed known variance, and establish sufficient sketch sizes given the problem dimensions.