Limit Theorem on Convergence to the Local Time of a Brownian Bridge

An integer random walk $$\{S_{i},\,i\geq 0\}$$ 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.