Set-indexed and multiple sums in high dimensions

TL;DR

This paper studies the convergence of set-indexed and multiple sums of high-dimensional random vectors, identifying a generalized Wiener spiral as the limit.

Contribution

It introduces a new high-dimensional limit object for set-indexed sums, extending the classical Wiener spiral concept.

Findings

Set-indexed sums converge in probability as dimension grows.

The limit generalizes the Wiener spiral to higher dimensions.

High-dimensional sums exhibit a specific convergence behavior.

Abstract

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.