Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals

The paper’s main theorem establishes, for the AR(1) process X_{k+1}=βX_k+ε_{k+1} with P(ε=0)=p, P(ε=1−β)=q, that with a_n=β^{-n}, b_n=1 and k_n=⌊q^{-n}⌋, the maxima satisfy lim_{n→∞} P(a_n(M_{k_n}−b_n)≤x)=exp{−p(−x)^{log q/ log β} ν_{β,q}(log(−x))} for all x<0; the extremal index is θ=p . This relies on the explicit max-semistable tail representation of the marginal F_{β,p} and its dual F_{β,q} near 0, plus a direct block/association argument to handle dependence . The candidate solution derives the same limit: (i) it matches the exact marginal tail asymptotics from F_{β,q} to get k_n P(X_0>1+xβ^n)→(−x)^{α}ν_{β,q}(log(−x)) with α=log q/log β; (ii) it identifies θ=p via a runs argument; and (iii) it applies standard EV theory for stationary sequences with extremal index to conclude the limit law. The paper’s proof and the model’s proof differ in technique (block/association versus O’Brien’s runs + D/D′-style conditions), but they agree on the statement and the key mechanisms (tail modulation and clustering), and no substantive conflict was found.