Particle Filter Concentration for Time-Varying Random Matrix Spectra

The work by Neeman et al. (2023) on concentration for sums of Markov-dependent matrices is a huge step forward, but it's still fundamentally static. It gives you a bound for a fixed sum. But many real-world problems, like dynamic networks or time-series analysis, involve matrices whose entries change over time. Now, consider the work by Moral et al. (2011) on concentration for Feynman-Kac particle processes. They developed incredible tools for understanding the concentration of complex, evolving systems. My idea is to synthesize these two fields. We can model the empirical spectral distribution of a time-varying random matrix as an interacting particle system, where each particle represents an eigenvalue trajectory. The rich concentration theory for particle filters could then be adapted to provide dynamic, time-dependent bounds on the entire spectrum. Instead of a single Hoeffding-type bound, you'd get a process that tells you how the concentration of the eigenvalues evolves. This is a paradigm shift from "here is a bound" to "here is how the bound behaves over time." It builds on the dependency structure from Neeman et al. but uses the powerful, dynamic techniques from Moral et al., opening up new possibilities for analyzing the stability of evolving systems in control theory, finance, and network science.

References:

1. Concentration inequalities for sums of Markov-dependent random matrices. Joe Neeman, Bobby Shi, Rachel A. Ward (2023). Information and Inference A Journal of the IMA.
2. On the Concentration Properties of Interacting Particle Processes. P. Moral, Peng Hu, Liming Wu (2011). Found. Trends Mach. Learn..