An estimate about multiple stochastic integrals with respect to a normalized empirical measure
| dc.creator | Major, Peter | |
| dc.date | 2003-10-20 | |
| dc.date.accessioned | 2026-07-07T05:02:07Z | |
| dc.date.available | 2026-07-07T05:02:07Z | |
| dc.description | Let a sequence of iid. random variables $ξ_1,...,ξ_n$ be given on a measurable space $(X,\cal X)$ with distribution $μ$ together with a function $f(x_1,...,x_k)$ on the product space $(X^k,{\cal X}^k)$. Let $μ_n$ denote the empirical measure defined by these random variables and consider the random integral $$ J_{n,k}(f)={n^{k/2}\over{k!}}\int' f(u_1,...,u_k) (μ_n(du_1)-μ(du_1))...(μ_n(du_k)-μ(du_k)), $$ where prime means that the diagonals are omitted from the domain of integration. In this work a good bound is given on the probability $P(|J_{n,k}(f)|>x)$ for all $x>0$. This result shows that the tail behaviour of the distribution funtcion of the random integral $J_{n,k}(f)$ 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 $J_{n,k}(f)$ 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. | |
| dc.identifier | https://arxiv.org/abs/math/0310323 | |
| dc.identifier | http://arxiv.org/abs/math/0310323 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68931 | |
| dc.subject | Probability | |
| dc.title | An estimate about multiple stochastic integrals with respect to a normalized empirical measure | |
| dc.type | text |