Moderate deviations for degenerate \(U\)-processes.
\(U\)-processes of order \(m\) defined on a probability space \((S,\mathcal S,\mu )\) and indexed by a collection of measurable functions \(\mathcal H\) are considered in case that each function \(h\in \mathcal H\) is completely degenerate (\(\mu \)-canonical). Sufficient conditions on \(\mathcal H \) are established under which the moderate \(U\)-process of rank \(r,\;2\leq r \leq m\), satisfies the moderate deviation principle (MDP). Especially, for \(\mathcal H\) to be Vapnik-Červonenkis subgraph class of function, the MDP is established under exponential moment condition on the envelope of \(\mathcal H .\) Some new Bernstein-type inequalities for \(U\)-processes are proved by using randomization and decoupling technique. Results on MDP for \(V\)-processes are also presented.
scientific article; zbMATH DE number 2098078
scientific article