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 Fr\'echet-type extreme-value law and quantitative error estimates. We prove a single explicit polynomial Kolmogorov bound that is uniform over the full level range $y>0$. More precisely, $\sup_{y>0}\left|γ_\vartheta(M_n\le ny)-\exp(-c_\vartheta/y)\right| \le C_\vartheta\frac{(\log n)^2}{\sqrt n}$. Here $c_\vartheta=1/\log(1+\vartheta^2)$. On every fixed range $y\ge\delta>0$ the same block argument yields the sharper rate $O_{\vartheta,\delta}((\log n/n)^{1/2})$. The proof uses only the explicit marginal tail and the exponential $\psi$-mixing estimate of the digit process; the global bound is obtained by pushing the block comparison to levels where the effective Poisson parameter is of order $\log n$ and then closing the remaining small-level range by monotonicity.