A functional limit convergence towards brownian excursion

We consider a random walk $S$ in the domain of attraction of a standard normal law $Z$, $\mathit{\text{ie}}$ there exists a positive sequence $a_n$ such that $S_n\mathrm{/}{a}_n$ converges in law towards $Z$. The main result of this note is that the rescaled process $(S_{\lfloor nt\rfloor }\mathrm{/}{a}_n,t\ge 0)$ conditioned to stay non-negative, to start and to come back $\mathit{\text{near}}\mathit{\text{the}}\mathit{\text{origin}}$ converges in law towards the normalized brownian excursion.