Upper bounds on product and multiplier empirical processes

We study two empirical processes of special structure: firstly, the centred multiplier process indexed by a class F, f→|∑i=1N(ξif(Xi)−Eξf)|, where the i.i.d. multipliers (ξi)i=1N need not be independent of (Xi)i=1N, and secondly, (f,h)→|∑i=1N(f(Xi)h(Xi)−Efh)|, the centred product process indexed by the classes F and H.We use chaining methods to obtain high probability upper bounds on the suprema of the two processes using a natural variation of Talagrand's γ-functionals.