  In this work we consider loop-erased random walk (LERW) and its scaling limit
in three dimensions, and prove that 3D LERW parametrized by renormalized length
converges to its scaling limit parametrized by some suitable measure with
respect to the uniform convergence topology in the lattice size scaling limit.
Our result greatly improves the work (Acta Math. 199(1):29-152) of Gady Kozma
which establishes the weak convergence of the rescaled trace of 3D LERW towards
a random compact set with respect to the Hausdorff distance.
