Functional limit theorem for the local time of stopped random walk

Abstract

AbstractInteger random walk {Sn, n ≥ 0} with zero drift and finite variance σ2 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 ⌊uσ$\begin{array}{} \displaystyle \sqrt{n} \end{array}$⌋ by this walk conditioned on T > n, the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.