# When products of projections diverge

## TL;DR

This paper explores the relationship between the convergence and divergence of cyclic and random products of projections in Hilbert spaces, providing a characterization of when random products converge based on geometric and combinatorial structures.

## Contribution

It establishes a parallel to known cyclic projection results by characterizing families of subspaces for which all random products converge.

## Key findings

- Cyclic projections always converge in norm.
- Random projections can diverge even when cyclic ones converge.
- The paper provides a geometric and combinatorial characterization of convergence for random products.

## Abstract

Slow convergence of cyclic projections implies divergence of random projections and vice versa. Let $L_1,L_2,\dots,L_K$ be a family of $K$ closed subspaces of a Hilbert space. It is well known that although the cyclic product of the orthogonal projections on these spaces always converges in norm, random products might diverge. Moreover, in the cyclic case there is a dichotomy: the convergence is fast if and only if $L_1^{\perp}+\dots+L_K^{\perp}$ is closed; otherwise the convergence is arbitrarily slow. We prove a parallel to this result concerning random products: we characterize those families $L_1,\dots,L_K$ for which all random products converge using their geometric and combinatorial structure.