$n^{-1/2}d_n(x) \phi(x)$ in (\[eq1.2\])) that reflects the “continuity correction” needed to approximate the discontinuous distribution of $T=\{S-E(S)\}/({\operatorname{var}}S)^{1/2}$ by a continuous normal distribution.

When exploring this problem, we suppose that the sequence of values of $n$ is strictly increasing. Further, we assume that $$\label{eq2.4} \min_{1\leq j\leq k} \liminf_{n{{\to}\infty}} (n_j/n)>0.$$ In Theorem \[the1\], below, we fix both $k$ and the distributions of $X_{j1}$, for $1\leq j\leq k$. This means that $e_1,\ldots,e_k$ are fixed too. However, for each $n$ we consider there to be a potentially new sequence of values $n_1,\ldots,n_k$. In particular, the ratios $n_{j_1}/n_{j_2}$ can change considerably from one choice of $n$ to another, although, in view of (\[eq2.4\]), $n_{j_1}/n_{j_2}$ is bounded away from zero and infinity as $n{{\to}\infty}$.

In the first part of Theorem \[the1\], below, we also impose the following condition on at least one of the ratios $\rho_{j_1j_2}=(e_{j_2}n_{j_1})/(e_{j_1}n_{j_2})$: $$\label{eq2.5} \mbox{for each integer $\ell\geq1$}, \qquad n^{1/2}|\sin(
\ell\rho_{j_1j_2}\pi)|{{\to}\infty}$$ as $n{{\to}\infty}$.

\[the1\] Assume that $E|X_{j1}|^3<\infty$ for $j=1,\ldots,k$; that $X_{j1}$ is distributed on a lattice $x_j+\nu e_j$, for integers $\nu
$, where $e_j$ is the maximal lattice edge width; and that (\[eq2.4\]) holds. () If, for some pair $j_1,j_2$ with $1\leq j_1<j_2\leq k$, $\rho_{j_1j_2}$ satisfies (\[eq2.5\]), then the one-term Edgeworth expansion at 