  In this paper we investigate the asymptotic properties of the wait-first and
jump-first L\'evy walk with rest, which is a generalization of standard
jump-first and jump-first L\'evy walk that assumes each waiting time in the
model is a sum of two positive random variables. We investigate the asymptotic
properties of the theses new-type waiting times. Next we use the previous
results of this paper together with continuous mapping approach to establish
the main result, which is a functional convergence in Skorokhod $\mathbb{J}_1$
topology for the L\'evy walks with rests.
