Many scientists try to tell whether a complex system—an ecosystem, a brain circuit, or a financial market—is operating near the threshold where a small shock can spread through the whole system. This paper asks a simple question: from a finite, noisy recording, how precisely can we estimate the system’s distance to that threshold? The authors study this using the multivariate Ornstein–Uhlenbeck process, a standard linear model for how small fluctuations relax back to a stable state.

The authors show that only three things control how well you can estimate the distance to instability. The first is an effective measurement budget, which depends not just on how many samples you have but on the system size (the number of variables) and the time between samples. The second is the signal-to-noise ratio, meaning how strong the deterministic interactions are compared with random fluctuations. The third is the distance to criticality itself, defined here as the smallest real part of the eigenvalues of the interaction matrix: r = min Re[λi(A)]. As r gets small, the slowest mode of the system “softens,” making fluctuations larger and correlations last longer. Those two effects both make estimation harder.

A key consequence is that temporal correlations near instability cut down the number of effectively independent observations far below the raw sample count. If the effective sample count drops below the system dimension, the empirical covariance matrix becomes singular and inference breaks down completely, even when the nominal data volume looks adequate. The authors identify three practical thresholds: a breakdown threshold where inference fails, a stability-diagnosis threshold below which one cannot reliably tell whether the system is stable or unstable, and a resolution threshold where the distance to instability can be estimated to within 10% relative precision. They also predict an optimal sampling interval, which grows large as the system approaches criticality.