  In this paper, we revisit the backward Euler method for numerical
approximations of random periodic solutions of semilinear SDEs with additive
noise. Improved $L^{p}$-estimates of the random periodic solutions of the
considered SDEs are obtained under a more relaxed condition compared to
literature. The backward Euler scheme is proved to converge with an order one
in the mean square sense, which also improves the existing order-half
convergence. Numerical examples are presented to verify our theoretical
analysis.
