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 -> vertical bar Sigma(N)(i=1) (xi(i) f(X-i) - Epsilon xi f) vertical bar, 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) -> vertical bar Sigma(N)(i=1) (f(X-i)h(X-i) -Epsilon fh) vertical bar, 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. (C) 2016 Published by Elsevier B.V.