  We consider in this work the convergence of Random Batch Method proposed in
our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] for interacting
particles to the case of disparate species and weights. We show that the strong
error is of $O(\sqrt{\tau})$ while the weak error is of $O(\tau)$ where $\tau$
is the time step between two random divisions of batches. Both types of
convergence are uniform in $N$, the number of particles. The proof of strong
convergence follows closely the proof in [Jin et al., J. Comput. Phys., 400(1),
2020] for indistinguishable particles, but there are still some differences:
since there is no exchangeability now, we have to use a certain weighted
average of the errors; some refined auxiliary lemmas have to be proved compared
with our previous work. To show that the weak convergence of empirical measure
is uniform in $N$, certain sharp estimates for the derivatives of the backward
equations have been used. The weak convergence analysis is also illustrating
for the convergence of Random Batch Method for $N$-body Liouville equations.
