Geodesic
Functional limit theorem for the local time of stopped random walk
Diskretnaya Matematika, Tome 31 (2019) no. 1, pp. 7-20

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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.
Keywords: conditioned Brownian motions, local time of conditioned Brownian motions, functional limit theorems. 
@article{DM_2019_31_1_a1,
     author = {V. I. Afanasyev},
     title = {Functional limit theorem for the local time of stopped random walk},
     journal = {Diskretnaya Matematika},
     pages = {7--20},
     publisher = {mathdoc},
     volume = {31},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2019_31_1_a1/}
}
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V. I. Afanasyev. Functional limit theorem for the local time of stopped random walk. Diskretnaya Matematika, Tome 31 (2019) no. 1, pp. 7-20. http://geodesic.mathdoc.fr/item/DM_2019_31_1_a1/