  Let a sequence of iid. random variables $\xi_1,...,\xi_n$ be given on a
measurable space $(X,\cal X)$ with distribution $\mu$ together with a function
$f(x_1,...,x_k)$ on the product space $(X^k,{\cal X}^k)$. Let $\mu_n$ denote
the empirical measure defined by these random variables and consider the random
integral $$ J_{n,k}(f)={{n^{k/2}}\over{k!}}\int' f(u_1,...,u_k)
(\mu_n(du_1)-\mu(du_1))...(\mu_n(du_k)-\mu(du_k)), $$ 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.
