On the Local Time of a Stopped Random Walk Attaining a High Level

An integer-valued random walk $\{S_i,\, i\geq 0\}$ with zero drift and finite variance $\sigma ^2$ stopped at the time $T$ of the first hit of the semiaxis $(-\infty ,0]$ is considered. For the random process defined for a variable $u>0$ as the number of visits of this walk to the state $\lfloor un\rfloor $ and conditioned on the event $\max _{1\leq i\leq T}S_i>n$, a functional limit theorem on its convergence to the local time of the Brownian high jump is proved.