A Uniform Polynomial Rate for the Extreme-Value Law of ϑ-Expansion Digits

Source abstract

Fix $\vartheta^2=1/m$, $m\in\mathbb{N}$, and let $M_n$ be the maximum of the first $n$ digits of the stationary $\vartheta$-expansion process under its invariant measure. Sebe, Lascu and Selmi established the corresponding Fréchet-type extreme-value law together with quantitative error estimates. We prove a single explicit polynomial Kolmogorov bound that is uniform over the full level range $y>0$: $\sup_{y>0}\left|\gamma_\vartheta(M_n\le ny)-\exp(-c_\vartheta/y)\right| \le C_\vartheta\frac{(\log n)^2}{\sqrt n}.$ Here $c_\vartheta=1/\log(1+\vartheta^2)$. On every fixed range $y\ge\delta>0$ the rate improves to $O_{\vartheta,\delta}((\log n/n)^{1/2})$. As an intermediate step we isolate an elementary blocking estimate for exponentially $\psi$-mixing stationary sequences which remains uniform while the Poisson parameter grows as $O(\log n)$. For the $\vartheta$-digit process, the exact marginal tail allows us to reach the boundary scale $y\asymp1/\log n$; monotonicity then closes the remaining small-level range.