  The exact expression for the probability density $p_{_N}(x)$ for sums of a
finite number $N$ of random independent terms is obtained. It is shown that the
very tail of $p_{_N}(x)$ has a Gaussian form if and only if all the random
terms are distributed according to the Gauss Law. In all other cases the tail
for $p_{_N}(x)$ differs from the Gaussian. If the variances of random terms
diverge the non-Gaussian tail is related to a Levy distribution for
$p_{_N}(x)$. However, the tail is not Gaussian even if the variances are
finite. In the latter case $p_{_N}(x)$ has two different asymptotics. At small
and moderate values of $x$ the distribution is Gaussian. At large $x$ the
non-Gaussian tail arises. The crossover between the two asymptotics occurs at
$x$ proportional to $N$. For this reason the non-Gaussian tail exists at finite
$N$ only. In the limit $N$ tends to infinity the origin of the tail is shifted
to infinity, i. e., the tail vanishes. Depending on the particular type of the
distribution of the random terms the non-Gaussian tail may decay either slower
than the Gaussian, or faster than it. A number of particular examples is
discussed in detail.
