Weak convergence from projected laws on a positive-measure set of directions

TL;DR

This paper shows that weak convergence of probability measures in  can be deduced from convergence along a positive-measure set of directions, given moment-determinacy of projected laws.

Contribution

It establishes that convergence on a positive-measure set of directions suffices for weak convergence, under moment-determinacy, simplifying the classical Crame9r-Wold characterization.

Findings

Weak convergence follows from projected convergence on a positive-measure set of directions.

Sampling a random direction almost surely yields weak convergence from projected laws.

Moment-determinacy of projected laws is key to the main result.

Abstract

The Cram\'er-Wold device characterises weak convergence of probability measures on $\mathbb{R}^d$ through convergence of all one-dimensional projected laws. We prove that, if the target projected laws are moment-determinate for surface-almost every direction, then weak convergence already follows from projected convergence on a positive-measure set of directions. This yields a simple probabilistic interpretation: if one samples a direction at random from any distribution on the sphere that is absolutely continuous with respect to surface measure, then, with probability one, convergence of the projected law along the sampled direction already forces global weak convergence under the same moment-determinacy assumption.