Limit theorem on convergence to the local time of a Brownian excursion - Afanasyev, Valeriy I. (Journal of the Belarusian State University. Mathematics and Informatics, 2026)
An integer random walk Si, i ≥ 0 with zero drift and finite variance σ2 is investigated. For a random pro-cess that assigns, to a variable u > 0, the number of hits of the specified random walk into the state \[uσ n floor \] up to time n and is considered under the condition that S1 > 0, …, Sn – 1 > 0, Sn ≤ 0, a functional limit theorem concerning convergence in distribution of the process to the local time of a Brownian excursion is proved.