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Integer random walk { S n  ,  n  ≥ 0} with zero drift and finite variance  σ  2  stopped at the moment  T  of the first visit to the half axis (-∞, 0] is considered. For the random process which associates the variable  u  ≥ 0 with the number of visits the state ⌊ uσn$\begin{array}{} \displaystyle \sqrt{n} \end{array}$   ⌋ by this walk conditioned on  T  &gt;  n , the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.

Functional limit theorem for the local time of stopped random walk