arXiv subjects
Publications and source records attributed to Bochen Jin.
We consider multiple and set-indexed sums of random vectors taking values in Euclidean space of growing dimension. It is shown that, when viewed as finite metric spaces, the sets of values of such sums converge in probability. The limit is identified as a generalisation of the Wiener spiral, which appears as the high-dimensional limit of single-index sums.
We consider $n$ independent random points uniformly distributed in the $d_n$-dimensional unit cube and study Pareto points, that is, points that do not coordinatewise dominate any other point. We identify the critical growth rate of $d_n$ at which a phase transition occurs: below this threshold, the number of non-Pareto points diverges in probability, whereas above it there are asymptotically no such points. At criticality, the number of non-Pareto points converges in distribution to a Poisson random variable. We further describe their asymptotic spatial distribution in terms of convergence of random point measures. We also investigate points that dominate exactly $r$ other points and establish analogous phase transitions. For $r=1$, the critical dimension is the same as for non-Pareto points, whereas for every fixed $r\geq 2$ it is different, but, surprisingly, common to all such $r$.
We prove the limit theorem for paths of random walks with $n$ steps in $\mathbb{R}^d$ as $n$ and $d$ both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the $\ell_p$-metric for $p\in[1,\infty)$. Under the assumptions that all components of each step are uncorrelated, centered, have finite $2p$-th moments, and are identically distributed, we show that such random metric space converges in probability to a deterministic limit space with respect to the Gromov-Hausdorff distance. This result generalises earlier work by Kabluchko and Marynych for $p=2$.
We consider extremal processes and random walks generated by heavy-tailed random vectors taking values in $\mathbb{R}^d$ endowed with the $\ell_p$ metric. We establish limit theorems for the associated paths in the triangular array setting when both the number of steps $n$ and the dimension $d$ grow to infinity. It is shown that it is possible to transform the paths by suitable isometries of $\ell_p$ such that the transformed paths converge in distribution and to identify the limit in terms of a Poisson cluster process. These results also imply the convergence in distribution of the paths viewed as finite metric spaces in the space of metric spaces equipped with the Gromov-Hausdorff metric. Furthermore, we prove convergence in distribution of the transformed paths in the space of counting measures on the line equipped with a Hausdorff metric induced by a suitable $\ell_p$-type distance between counting measures.
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 $\alpha \in (0,1)$, the limiting metric space has a random metric derived from the bridge variant of a subordinator.