You can now prove a high-dimensional object is stable just by looking at its shadows from a few random angles.
April 16, 2026
Original Paper
Weak convergence from projected laws on a positive-measure set of directions
arXiv · 2604.10310
The Takeaway
In statistics and physics, understanding high-dimensional data is a nightmare because you usually have to look at it from every single possible direction to be sure it's behaving. This is known as the Cramér-Wold device, and it’s essentially the mathematical equivalent of needing to take an infinite number of photos of a statue to prove it exists. This paper throws that rule out the window, proving that a 'positive-measure' set—basically just a random, decent sample of directions—is enough to guarantee the whole thing is converging. It’s a massive shortcut for anyone dealing with complex systems, from neural networks to particle physics. It means we can stop worrying about the infinite 'missing angles' and trust that a small, well-chosen glimpse gives us the whole truth.
From the abstract
The Cramér-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 abso