Random walks in the high-dimensional limit. II: The crinkled subordinator (Q6615476)

The authors consider a sequence of random variables defined on a common probability space and taking values in \(\mathbb{R}^d\). The random walk is created in a standard way as the sum of these random variables and the number of terms in the random walk depends on \(d\). The piecewise-linear interpolation is obtained by joining the consecutive points of the random walk by line segments. Also, let \(\ell^2\) be the infinite-dimensional (real) Hilbert space of square-summable sequences endowed with the standard Hilbert norm. Then every representative of the random walk can be regarded as a continuous piecewise-linear curve in \(\ell^2\) starting at the origin and living in the finite-dimensional subspace \(\mathbb{R}^d\). The paper is devoted to finding an answer to the question: How does the created curve (after an appropriate renormalization and up to isometries in \(\ell^2\)) look like when \(d\) and therefore, the number of terms in the random walk tend to infinity? Does it approach some deterministic or random curve in \(\ell^2\)? To answer this question, a crinkled subordinator is considered that is an \(\ell^2\)-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. It is proved that the path of the considered random walk with \(n\) independent identically distributed steps having heavy-tailed distribution of the radial components and asymptotically orthogonal angular components, converges in distribution in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as \(d\) and \(n\) tend to infinity.\N\NFor Part I see [the first two authors, Ann. Inst. Henri Poincaré, Probab. Stat. 60, No. 4, 2945--2974 (2024; Zbl 07967667)].