ENRICHED FUNCTIONAL LIMIT THEOREMS FOR DYNAMICAL SYSTEMS

The paper proves an enriched functional limit theorem in the new space F′: suitably normalized partial sums with clustered extremes converge to an α-stable Lévy process V, with jump times given by a Poisson process and excursions encoding intra-cluster paths (Theorem 2.5). The proof proceeds via complete convergence of a decorated cluster point process to a Poisson cluster process with intensity θ Leb×Leb and cluster-shape marks, followed by a continuous mapping to the path-and-excursion space (Theorem 3.16 and ensuing arguments). The candidate solution reproduces this structure: it constructs the same Poisson cluster limit, applies a continuous mapping (via Whitt’s E/decorated-space ideas) to obtain convergence in F′, handles small-jump/centering and α=1 via the same integrability/log-moment assumptions, and identifies V by the same Poisson series/LePage representation. Minor differences are expository (use of tail/spectral-tail process vs. the paper’s piling process, and where continuity is verified), but they are mathematically compatible with the paper’s framework. Key elements—Poisson cluster limit with intensity θ Leb×Leb and marks Q̃, the small-jump control (2.21), integrability/log-moment conditions (2.22)–(2.23), and excursion parametrization t↦∑_{j≤⌊tan(π(t−1/2))⌋}Q_j—match the paper’s statements and proofs. Hence both are correct and substantially the same proof at the level of ideas and objects. Supporting citations include the statement of Theorem 2.5 with the explicit series/compensated forms and excursions, the Poisson cluster point process construction and convergence (Theorem 3.16), and the small-jump and normalization details for the partial sums.