Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel

The paper’s Theorem 2.7 states precisely the two claims the candidate proves: (i) asymptotic normality of the ORALS operator-regression estimator zi,M with covariance σ^2Δt A−1_{i,∞}, and (ii) consistency (up to a sign) and asymptotic normality of the first c-update and second a-update in the deterministic ALS stage, starting from any c0 with c*⊤c0 ≠ 0. The paper’s proof (Appendix A.3) proceeds by the same OLS-with-random-design/CLT/Slutsky route as the candidate for (i), and linearizes the normalization step and the ALS updates—introducing η and ε perturbations—to derive the same Gaussian limits in (ii). See the statement of Theorem 2.7 and its proof, which explicitly uses LLN/CLT across independent trajectories m, the kernel coercivity and uniform boundedness of the basis to ensure invertibility and tightness, and the same two-step ALS expansion culminating in the projection [I − uu⊤]Ξic* term (the candidate’s derivation matches the paper’s displays (A.22)–(A.26)). Coercivity and invertibility are handled in Proposition 2.6 under Assumption 2.5 and Definition 2.4, exactly as invoked by the candidate. No substantive discrepancies were found; the candidate’s solution is a clean paraphrase with the same logical steps and limiting distributions.