obtains if $\rho_{j_1j_2}=1+{\varepsilon}_{j_1j_2}$, where ${\varepsilon}_{j_1j_2}={\varepsilon}_{j_1j_2}(n)$, which can be either positive or negative, converges to zero strictly more slowly than $n^{1/2}$: $$\label{eq2.6} {\varepsilon}_{j_1j_2}{\to}0, \qquad n^{1/2}|
{\varepsilon}_{j_1j_2}| {{\to}\infty}.$$ In this case, for each fixed, positive integer $\ell$, $$\sin(\ell\rho_{j_1j_2}\pi)=\sin(\ell\pi)+\ell\pi{\varepsilon}_{j_1j_2} \cos(
\ell\pi)+O \bigl({\varepsilon}_{j_1j_2}^2 \bigr),$$ from which it follows that $$\label{eq2.7} \bigl|\sin(\ell\rho_{j_1j_2}\pi)\bigr| \sim \cases{ \ell
\pi| {\varepsilon}_{j_1j_2}| & if $\ell$ is an even integer
\cr
1 & if $\ell$ is an
odd integer,}$$ where $a_n\sim b_n$ means that the ratio $a_n/b_n$ converges to 1. Assumption (\[eq2.5\]) follows from (\[eq2.6\]) and (\[eq2.7\]).

A similar argument can be used to prove that if $\rho_{j_1j_2}=\rho_0+{\varepsilon}_{j_1j_2}$, where $\rho_0$ is a fixed rational number and ${\varepsilon}_{j_1j_2}={\varepsilon}_{j_1j_2}(n)$ satisfies (\[eq2.6\]), then (\[eq2.5\]) is true. (The case $\rho_0=0$ is excluded by (\[eq2.4\]).) These examples make it clear that there is not a great deal of latitude in the assumption, imposed in part (ii) of Theorem \[the1\], that each $\rho_{j_1j_2}$ should equal a fixed rational number. In particular, for (\[eq2.5\]) to fail it is not sufficient that each $\rho_{j_1j_2}$ converge to a rational.

Refinement of bound on remainder term in Edgeworth expansions {#sec2.3}
-------------------------------------------------------------

In Section \[sec2.1\], we showed that, if (\[eq2.5\]) holds, the discontinuous term of size $n^{-1/2}$, in expansions such as (\[eq1.2\]), 