  It is well known (Donsker's Invariance Principle) that the random walk
converges to Brownian motion by scaling. In this paper, we will prove that the
scaled local time of the $(1,L)-$random walk converges to that of the Brownian
motion. The results was proved by Rogers (1984) in the case $L=1$. Our proof is
based on the intrinsic multiple branching structure within the $(1,L)-$random
walk revealed by Hong and Wang (2013).
