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              <p>&lt;p&gt;We prove that when a sequence of Lévy processes &lt;em&gt;X&lt;sup&gt;(n)&lt;/sup&gt;&lt;/em&gt; or a normed sequence of random walks &lt;em&gt;S&lt;sup&gt;(n)&lt;/sup&gt;&lt;/em&gt; converges a.s. on the Skorokhod space toward a Lévy process &lt;em&gt;X&lt;/em&gt;, the sequence &lt;em&gt;L&lt;sup&gt;(n)&lt;/sup&gt;&lt;/em&gt; of local times at the supremum of &lt;em&gt;X&lt;sup&gt;(n)&lt;/sup&gt;&lt;/em&gt;&lt;sup&gt; &lt;/sup&gt;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 toward the ladder processes of &lt;em&gt;X&lt;/em&gt;. As an application, we show that in general, the sequence &lt;em&gt;S&lt;sup&gt;(n)&lt;/sup&gt;&lt;/em&gt; conditioned to stay positive converges weakly, jointly with its local time at the future minimum, toward the corresponding functional for the limiting process &lt;em&gt;X&lt;/em&gt;. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.&lt;/p&gt;</p>
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