Abstract
We consider a random walk \(\left\{ S_{n},\text { }n\ge 0\right\}\) with zero drift and finite variance \(\sigma ^{2}\). Let T be the first hitting time of the semi-axis \(\left( -\infty ,0\right]\) by this random walk. For the random process \(\left\{ S_{\left\lfloor nt\right\rfloor }/\sigma \sqrt{n}{\small ,}\text { }t\in \left[ 0,1\right] \right\}\), considered under the condition that \(T=n\), a functional limit theorem on its convergence to the Brownian excursion, as \(n\rightarrow \infty\), is proved.
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29 March 2023
A Correction to this paper has been published: https://doi.org/10.1007/s10958-023-06325-0
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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. 075-15-2022-265). The corresponding author states that there is no conflict of interest. The manuscript has no associated data.
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Afanasyev, V.I. LOCAL INVARIANCE PRINCIPLE FOR A RANDOM WALK WITH ZERO DRIFT. J Math Sci 266, 850–868 (2022). https://doi.org/10.1007/s10958-022-06145-8
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DOI: https://doi.org/10.1007/s10958-022-06145-8