Year: 2025
Type: preprint
Abstract: We investigate the limiting behaviour of the path of random bridges treated as random sets in $\mathbb{R}^{d}$ with the Euclidean metric and the dimension $d$ increasing to infinity. The main result states that, in the square integrable case, the limit (in the Gromov-Hausdorff sense) is deterministic, namely, it is $[0,1]$ equipped with the pseudo-metric $\sqrt{|t-s|(1-|t-s|)}$. We also show that, in the heavy-tailed case with summands regularly varying of order $α\in (0,1)$, the limiting metric space has a random metric derived from the bridge variant of a subordinator.
Source: arXiv (Cornell University)
Author: Jin, Bochen
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979298568
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