  We establish the discrete approximation to Brownian motion with varying
dimension (BMVD in abbreviation) by random walks. The setting is very similar
to that in [11], but here we use a different method allowing us to get rid the
restrictions in [11] (or [3]) that the underlying state space has to be
bounded, and that the initial distribution of the limiting continuous process
has to be its invariant distribution. The approach in this paper is that we
first obtain heat kernel upper bounds for the approximating random walks that
are uniform in their mesh size, by establishing a Nash-type inequality based on
their Dirichlet form characterization. Using the heat kernel upper bound, we
then show the tightness of the approximating random walks by delicate analysis.
