Random Bridges in Spaces of Growing Dimension

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 $\sqrt{\mathrm{\mid }t-s\mathrm{\mid }(1-\mathrm{\mid }t-s\mathrm{\mid })}$. We also show that, in the heavy-tailed case with summands regularly varying of order $\alpha \in (0,1)$, the limiting metric space has a random metric derived from the bridge variant of a subordinator.