Birkhoff Sum Convergence of Fréchet Observables to Stable Laws for Gibbs-Markov Systems and Applications

The paper proves small-jumps negligibility (Theorem 8.1) for α in (1,2) by verifying a Davis–Hsing-type mixing/covariance condition using BV decay of correlations, then combines this with point-process limits for Fréchet observables to obtain stable laws (Corollary 8.3) and treats intermittent (LSV) maps via inducing, yielding the three-case theorem (Theorem 8.5) . The candidate solution reaches essentially the same conclusions: (A) small-jumps negligibility for α∈(1,2) via a martingale/Poisson-equation approach and Freedman’s inequality; (B) α-stable limits for φ(x)=d(x,x0)^{-1/α} on Gibbs–Markov maps; and (C) the LSV trichotomy with the same parameter regimes. However, the candidate’s proof strategy differs and has technical gaps: it sketches a reverse-martingale decomposition based on (I−L)h=f with an extra remainder term and asserts a variance control E[V_{n,ε}]≲nE[f_{n,ε}^2] without justification, and it does not account for Poisson clustering near periodic points (addressed in the paper via point-process limits of Freitas–Freitas–Magalhães and Davis–Hsing) . Despite these gaps in the model’s outline, the results it states match the paper’s theorems and parameter thresholds (including the cancellation regime bound α<1+1/γ^2−1/γ) .