2003/10/20 by Péter Major, Peter Major, Major, Peter
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #advanced mathematical theories #math.PR
paper · pdf · doi:10.48550/arxiv.math/0310323
arxiv created 2003/10/20 · openalex publication_date 2003/10/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let a sequence of iid. random variables ξ1,...,ξn be given on a measurable space (X,\cal X) with distribution μ together with a function f(x1,...,xk) on the product space (Xk,\cal Xk). Let μn denote the empirical measure defined by these random variables and consider the random integral Jn,k(f)=nk/2\overk!∫' f(u1,...,uk) (μn(du1)-μ(du1))...(μn(duk)-μ(duk)), where prime means that the diagonals are omitted from the domain of integration. In this work a good bound is given on the probability P(|Jn,k(f)|>x) for all x>0. This result shows that the tail behaviour of the distribution funtcion of the random integral Jn,k(f) and that of the integral of the function f with respect to a Gaussian random field show a similar behaviour. The proof is based on an adaptation of some methods of the theory of Wiener--Ito integrals. In particular, a sort of diagram formula is proved for the random integrals Jn,k(f) together with some of its important properties, a result which may be interesting in itself. The relation of this estimate to some results about U-statistics is also discussed.