Upper bounds on product and multiplier empirical processes

We study two empirical process of special structure: firstly, the centred multiplier process indexed by a class $F$, $f \to \left|\sum_{i=1}^N (\xi_i f(X_i) - \E \xi f)\right|$, where the i.i.d. multipliers $(\xi_i)_{i=1}^N$ need not be independent of $(X_i)_{i=1}^N$, and secondly, $(f,h) \to \left|\sum_{i=1}^N (f(X_i)h(X_i)-\E f h) \right|$, 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 $\gamma$-functionals.