An estimate about multiple stochastic integrals with respect to a normalized empirical measure

ISSN： 0081-6906

Source： Studia Scientiarum Mathematicarum Hungarica, Vol.42, Iss.3, 2005-08, pp. : 295-312

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Abstract

Let a sequence of iid. random variables $xi_1,dots,xi_n$ be given on a measurable space $(X,mathcal{X})$ with distribution $mu$ together with a function $f(x_1,dots,x_k)$ on the product space $(Xk,mathcal{X}k)$. Let $mu_n$ denote the empirical measure defined by these random variables and consider the random integral $$J_{n,k}(f)=frac{n{k/2}}{k!}int olimits' f(u_1,dots,u_k)sb(mu_n(,du_1)-mu(du_1)sb)dots sb(mu_n(,du_k)-mu(,du_k)sb),$$ where prime means that the diagonals are omitted from the domain of integration. A good bound is given on the probability $PB(b|J_{n,k}(f)b|>xB)$ for all $x>0$ which is similar to the estimate in the analogous problem we obtain by considering the Gaussian (multiple) Wiener--Ito integral of the function $f$. 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 paper to some results about $U$-statistics is also discussed.