  Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural
number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n
a_j}x) where the functions f_j are bounded, and a_j are integers. A factor of X
is characteristic for averaging schemes of length k (or k-characteristic) if
for any non zero distinct integers a_1,...,a_k, the limiting L^2(\mu) behavior
of the averages in (*) is unaltered if we first project the functions f_j onto
the factor. A factor of X is a k-universal characteristic factor (k-u.c.f)} if
it is a k-characteristic factor, and a factor of any k-characteristic factor.
We show that there exists a unique k-u.c.f, and it has a structure of a
(k-1)-step nilsystem, more specifically an inverse limit of (k-1)-step
nilflows. Using this we show that the averages in (*) converge in L^2(\mu).
This provides an alternative proof to the one given by Host and Kra in 2002.
