Geodesic
On the Local Time of a Stopped Random Walk Attaining a High Level
Informatics and Automation, Branching Processes and Related Topics, Tome 316 (2022), pp. 11-31

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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.
Keywords: conditional Brownian motion, local time, functional limit theorem. 
@article{TRSPY_2022_316_a2,
     author = {V. I. Afanasyev},
     title = {On the {Local} {Time} of a {Stopped} {Random} {Walk} {Attaining} a {High} {Level}},
     journal = {Informatics and Automation},
     pages = {11--31},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2022_316_a2/}
}
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V. I. Afanasyev. On the Local Time of a Stopped Random Walk Attaining a High Level. Informatics and Automation, Branching Processes and Related Topics, Tome 316 (2022), pp. 11-31. http://geodesic.mathdoc.fr/item/TRSPY_2022_316_a2/