A stochastic differential equation with graph-structured drift produces a steady-state covariance matrix satisfying a Lyapunov equation. But the covariance alone cannot recover the drift — multiple parameter configurations generate identical second-order statistics. The system is unidentifiable from its covariance.
Under non-Gaussianity, higher-order cumulants — third, fourth, and beyond — provide additional algebraic constraints that generically identify the drift matrix for any connected graph.
The reason is precise. Gaussian distributions have all cumulants of order three and above equal to zero. Those higher-order equations don't simplify — they collapse entirely, becoming 0 = 0. They carry no information. The cumulant equations exist for any distribution, but Gaussianity annihilates them. Non-Gaussianity makes the equations non-trivial, and these extra constraints overdetermine the system, resolving the ambiguity that covariance alone cannot.
This is not a marginal improvement. The jump from unidentifiable to generically identifiable happens the moment you move away from Gaussianity. The assumption that made everything tractable was simultaneously destroying the equations that would have let you recover the system's causal structure.
The practical complication: the resulting semiparametric estimator requires large sample sizes. The identifiability is theoretically generic but practically fragile. You can in principle recover what Gaussianity hid, but only with enough data to resolve the higher-order structure.
The deeper point is not about cumulants or Lyapunov equations specifically. It is about what convenient assumptions cost. Gaussianity is the most common simplifying assumption in statistics — it makes integrals tractable, distributions symmetric, everything analytically clean. But the convenience is purchased by destroying precisely the information that would distinguish one generating process from another. The nice distribution is the one that hides the most.