This article studies the theoretical properties of the numerical scheme for backward stochastic differential equations, extending the relevant results of Briand et al. with more general assumptions. To be more precise, the Brown motion will be approximated using the sum of a sequence of martingale differences or a sequence of i.i.d. Gaussian variables instead of the i.i.d. Bernoulli sequence. We cope with an adaptation problem of $Y^n$ by defining a new process ${\widehat{Y}}^n$; then, we can obtain the Donsker-type theorem for numerical solutions using a similar method to Briand et al.