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.