  We consider interference effects between p-wave resonance scattering
amplitude and background s-wave amplitude in low-energy neutron scattering from
a heavy nucleus which goes through the compound nucleus stage. The first effect
is in the difference between the forward and backward scattering cross
sections. Because of the chaotic nature of the compound states, this effect is
a random variable with zero mean. However, a statistical consideration shows
that the probability distribution of this effect does not obey the standard
central limit theorem. That is, the probability density for the effect averaged
over n resonances does not become a Gaussian distribution with the variance
decreasing as 1/sqrt(n) (``violation'' of the theorem!). We derive the
probability distribution of the effect and the limit distribution of the
average. It is found that the width of this distribution does not decrease with
the increase of n, i.e., fluctuations are not suppressed by averaging.
Furthermore, we consider the correlation between the neutron spin and the
scattering plane and find that this effect, although much smaller, shows
fluctuations which actually increase upon averaging over many measurements.
Limits of the effects due to finite resonance widths are also considered. In
the appendix we present a simple derivation of the limit theorem for the
average of random variables with infinite variances.
