  Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots,
n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic
means
  \begin{equation}
  \frac{1}{s_1\cdots
s_{n}}\sum_{j_1=0}^{s_1-1}\cdots\sum_{j_n=0}^{s_n-1}f\left(U_1^{j_1}\cdots
U_n^{j_n} x \right)
  \end{equation} as $\min_j s_j\to\infty $, for any function $f\in
L\log^{d-1}(X)$, where $d\le n$ is the rank of the transformations. The result
gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the
convergence of the means in a narrower class of functions $L\log^{n-1}(X)$.
