Limit theorem on convergence to the local time of a Brownian bridge

An integer random walk $\{S_{i},\,i\geqslant 0\}$ with zero drift and finite variance $\sigma^{2}$ is considered. For a random process that assigns, to a variable $u\in \mathbb{R}$, the number of hits of the specified walk into the state $\lfloor u\sigma\sqrt n\rfloor $ up to time $n$ and is considered under the condition that $S_{n}=0$, a functional limit theorem concerning convergence of the process to the local time of the Brownian bridge is proved.