Bounds for stochastic processes on product index spaces

In this paper we discuss the question how to bound supremum of a stochastic process with the index set of a product type. There is a tempting idea to approach the question by the analysis of the process on each of the marginal index spaces separately. However it turns out that we also need to study suitable partitions of the whole index space. We show what can be done in this direction and how to use the method to reprove some known results. In particular we observe that all known applications of the Bernoulli Theorem can be obtained in this way, moreover we use the shattering dimension to slightly extend the application to VC classes. We also show some application to the regularity of paths for processes which take values in vector spaces. Finally we give a short proof of the Mendelson-Paouris result on sums of squares for empirical processes.