  We investigate the statistics of the number $N(R,S)$ of lattice points, $n\in
\Z^2$, in a ``random'' annular domain $\Pi(R,w)=\,(R+w)A\,\setminus RA$, where
$R,w >0$. Here $A$ is a fixed convex set with smooth boundary and $w$ is chosen
so that the area of $\Pi (R,w)$ is $S$. The randomness comes from $R$ being
taken as random ( with a smooth denisity ) in some interval $[c_1T,c_2T]$,
$c_2>c_1>0$. We find that in the limit $T\to\infty $ the variance and
distribution of $\De N=N(R;S)-S$ depends strongly on how $S$ grows with $T$.
There is a saturation regime $S/T\to\infty$, as $T\to\infty$ in which the
fluctuations in $\Delta N$ coming from the two boundaries of $\Pi $, are
independent. Then there is a scaling regime, $S/T\to z$, $0<z<\infty $ in which
the distribution depends on $z$ in an almost periodic way going to a Gaussian
as $z\to\ 0$. The variance in this limit approaches $z$ for ``generic'' $A$ but
can be larger for ``degenerate'' cases. The former behavior is what one would
expect from the Poisson limit of a distribution for annuli of finite area.
