Let $T$ be an ergodic transformation of a nonatomic probability space,
$f$ an
$L_2$-function, and $K\geq1$ an integer. It is shown that there is
another
$L_2$-function $g$, such that the joint distribution of $T^ig$, $1\leq
i\leq K$,
is nearly
normal, and such that the corresponding inner products $(T^if,\,T^jf)$
and
$(T^ig,\,T^jg)$ are nearly the same for $1\leq i,\,j\leq K$. This
result can
be used to give a simpler and more transparent proof of an important
special
case of an earlier theorem [1], which was a refinement of
Bourgain's entropy
theorem [2].
[1] M. Akcoglu, M.D. Ha, and R.L. Jones, {\it
Sweeping-out properties of
operator sequences}\/,
Canad. J. Math. {\bf 49} (1997), 3--23.
[2] J. Bourgain, {\it Almost sure convergence and
bounded entropy\/}, Israel J. Math. {\bf 63} (1988), 79--97.