Functional limit theorem for the local time of stopped random walk

Integer random walk $\left\{ S_{n},\,n\geq 0\right\} $ with zero drift and finite variance $\sigma ^{2}$ stopped at the moment $T$ of the first visit to the half axis $\left( -\infty ,0\right] $ is considered. For the random process which associates the variable $u\geq 0$ with the number of visits the state $\left\lfloor u\sigma \sqrt{n}\right\rfloor $ by this walk conditioned on $T>n$, the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.