  We investigate error orders for integral limit approximations to traces of
products of Toeplitz matrices generated by integrable functions on $[-\pi,\pi]$
having some singularities at the origin. Even though a sharp error order of the
above approximation is derived in Theorem~2 of \cite{Lieberman-Phillips-2004},
its proof contains an inaccuracy as pointed out by
\cite{Ginovyan-Sahakyan-2013}. In the present paper, we reinvestigate the claim
given in Theorem~2 of \cite{Lieberman-Phillips-2004} and give an alternative
proof of their claim.
