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We consider regenerative processes with values in some general Polish space. We define their $\epsilon$-big excursions as excursions $e$ such that $\varphi \left(e\right)gt;\epsilon$, where $\varphi$ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $\epsilon$-big excursions and of their endpoints, for all $\epsilon$ in a set whose closure contains $0$. Finally, we provide...