  Assume that $m,s\in\mathbb N$, $m>1$, while $f$ is a polynomial with integer
coefficients, $\text{deg}~f>1$, $f^{(i)}$ is the $i$th iteration of the
polynomial $f$, $\kappa_n$ has a discrete uniform distribution on the set
$\{0,1,\ldots,m^n - 1\}$. We are going to prove that with $n$ tending to
infinity random vectors $\left(\frac{\kappa_n}{m^n},\frac{f(\kappa_n) \bmod
m^n}{m^n},\ldots,\frac{f^{(s - 1)}(\kappa_n) \bmod m^n}{m^n}\right)$ weakly
converge to a vector having a continuous uniform distribution in the
$s$-dimensional unit cube. Analogous results were obtained earlier only for
some classes of polynomials with $s\leqslant 3, \text{deg}~f = 2$.
  The mentioned vectors represent sequential pseudorandom numbers produced by a
polynomial congruential generator modulo $m^n$.
