  We examine the behavior of the number of $k$-term arithmetic progressions in
a random subset of $\mathbb{Z}/n\mathbb{Z}$. We prove that if a set is chosen
by including each element of $\mathbb{Z}/n\mathbb{Z}$ independently with
constant probability $p$, then the resulting distribution of $k$-term
arithmetic progressions in that set, while obeying a central limit theorem,
does not obey a local central limit theorem. The methods involve decomposing
the random variable into homogeneous degree $d$ polynomials with respect to the
Walsh/Fourier basis. Proving a suitable multivariate central limit theorem for
each component of the expansion gives the desired result.
