Abstract
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.
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Funding
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).
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Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 316, pp. 11–31 https://doi.org/10.4213/tm4215.
Translated by I. Nikitin
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Afanasyev, V.I. On the Local Time of a Stopped Random Walk Attaining a High Level. Proc. Steklov Inst. Math. 316, 5–25 (2022). https://doi.org/10.1134/S0081543822010035
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DOI: https://doi.org/10.1134/S0081543822010035