${\bar X}_j=n_j^{-1}\sum_iX_{ji}$, $\mu_j=E(X_{ji})$, ${\sigma}_j^2={\operatorname{var}}(X_{ji})$ and $$\label{eq2.1} S=\sum_{j=1}^k {\bar
X}_j.$$ The model (\[eq2.1\]) includes cases of apparently greater generality, for example where signed weights are incorporated in the series, since the absolute values of the weights can be incorporated into (\[eq2.1\]) by modifying the lattice edge widths, and negative signs can be addressed by reflecting the summand distributions.

Since third moments are finite then, if the distributions of $X_{11},\ldots,X_{k1}$ were to satisfy a smoothness condition, such as that of Cramér, we could express the distribution of $S$ in a one-term Edgeworth expansion: $$\label{eq2.2} P \biggl\{\frac{S-E(S)}{({\operatorname{var}}S)^{1/2}}\leq x \biggr\} =\Phi(x)+n^{-1/2}{
\frac{1}{6}}{\beta}\bigl(1-x^2 \bigr) \phi(x)+o
\bigl(n^{-1/2} \bigr) ,$$ where we take $n=n_1+\cdots+n_k$ to be the asymptotic parameter, and $$\label{eq2.3} {\beta}={\beta}(n)=\frac{n^{1/2}E(S-ES)^3}{({\operatorname{var}}S)^{3/2}} =\frac{n^{1/2}\sum_jn_j^{-2}E(X_{j1}-EX_{j1})^3}{(\sum_jn_j^{-1}{\operatorname{var}}X_{j1})^{3/2}}$$ is a measure of standardised skewness and, under our assumptions, is bounded as $n{{\to}\infty}$. Result (\[eq2.2\]) is a version of (\[eq1.1\]) in a particular case.

However, in general (\[eq2.2\]) does not hold in the lattice-valued case that we are considering. For example, if $k\geq1$ and the $X_{ji}$s, for all $i$ and $j$, have a common lattice distribution, then, as was made clear by Esseen [@Ess45], any expansion of the distribution of $X$ has to include a discontinuous term of size $n^{-1/2}$ (specifically, the term 