The distribution of short moving character sums

Sums of Dirichlet characters are one of the most studied objects in analytic number theory. In this talk I will describe some work on the distribution of
$$ \sum\limits_{x\,<\,n\,\leqslant\,x+H}\chi(n), $$
where $\chi$ is a non-principal character $\mod{q}$ and $x$ varies between $0$ and $q-1$. This problem was investigated by Davenport and Erdös, and more recently by Lamzouri and others. Lamzouri conjectured that provided
$$ H\to\infty, \quad\text{but}\quad H = o\left(\frac{q}{\log{q}}\right), $$
the sum should have a Gaussian limiting distribution. I will present some results that shed more light on this conjecture.