Escape Rate and Conditional Escape Rate from a Probabilistic Point of View

The paper’s main theorem explicitly proves that for a good neighbourhood system in a right φ-mixing setting with φ(k) ≤ Ck^{-p}, p>1, and with ∑ℓ ℓ α̂ℓ < ∞, the localized escape rate equals the extremal index α1 (Theorem A), using a block argument that reduces the problem to short-entry probabilities (Lemma 4.3) and then identifies the limit as α1 (Lemma 4.6) . By contrast, the model’s proof hinges on the set inclusions {τB>t+K} ⊂ {τU>t} ⊂ {τB>t} to assert ρ(U)=ρ(B); the left inclusion is generally false due to clustering of short returns, so the claimed equality of escape rates does not follow. The paper does not use such a step and instead derives the result via a careful block decomposition and mixing estimates .