  A $p$-adic Brownian motion is a continuous time stochastic process in a
$p$-adic state space that has a Vladimirov operator as its infinitesimal
generator. The current work shows that any such process is the scaling limit of
a discrete time random walk on a discrete group. Earlier work required the
exponent of the Vladimirov operator to be in $(1, \infty)$, and the convergence
was the weak convergence of probability measures on the Skorohod space of paths
on a compact time interval. The current approach simplifies the earlier
approach, allows for any positive exponent, eliminates the restriction to
compact time intervals, and establishes some moment estimates for the discrete
time processes that are of independent interest.
