On the tail behaviour of the distribution function of multiple stochastic integrals (Q1093248)

Let \(F_ n(u)\) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0,1]. Define ū\({}^ *_ n(u)=\sqrt{n}| F_ n(u)-u|\), and consider the integrals \[ I(t)=\int ^{t}_{0}\int ^{1}_{0}...\int ^{1}_{0}f(u_ 1,...,u_ s){\bar \mu}^ *_ n(du_ 1)...\mu ^ *_ n(du_ s), \] where f is a bounded measurable function. We give a good upper bound on the probability P(\(\underline{\sup}_{0\leq t\leq 1}| I(t)| \geq x)\). An analogous estimate is given for multiple integrals with respect to a Poisson process.