  We prove that when a sequence of L\'evy processes $X^{(n)}$ or a normed
sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward
a L\'evy process $X$, the sequence $L^{(n)}$ of local times at the supremum of
$X^{(n)}$ converges uniformly on compact sets in probability toward the local
time at the supremum of $X$. A consequence of this result is that the sequence
of (quadrivariate) ladder processes (both ascending and descending) converges
jointly in law towards the ladder processes of $X$. As an application, we show
that in general, the sequence $S^{(n)}$ conditioned to stay positive converges
weakly, jointly with its local time at the future minimum, towards the
corresponding functional for the limiting process $X$. From this we deduce an
invariance principle for the meander which extends known results for the case
of attraction to a stable law.
