Convergence to the local time of Brownian meander

Let $\left\{ S_{n},\;n\geq 0\right\}$ be integer-valued random walk with zero drift and variance $\sigma^2$. Let $\xi(k,n)$ be number of $t\in\{1,\ldots,n\}$ such that $S(t)=k$. For the sequence of random processes $\xi(\lfloor u\sigma \sqrt{n}\rfloor,n)$ considered under conditions $S_{1}>0,\ldots ,S_{n}>0$ a functional limit theorem on the convergence to the local time of Brownian meander is proved.