Effective Erd\H{o}s--Wintner for Cantor numeration systems via a trailing-window method

TL;DR

This paper introduces a trailing-window method to derive explicit Erd ext{o}s--Wintner bounds for Cantor numeration systems, improving convergence rates and providing a unified approach applicable to fixed-base and Cantor systems.

Contribution

It develops a simple trailing-window decomposition technique to obtain explicit probabilistic bounds for Cantor numeration systems, extending classical results to more general systems.

Findings

Recovered Delange's product in fixed-base case

Established explicit convergence bounds for Cantor systems

Provided a guide for choosing optimal regimes

Abstract

We prove explicit Erd\H{o}s--Wintner bounds for Cantor numeration systems via a simple trailing-window decomposition. We temporarily discard the last block of digits (the ``window'') and analyze the remaining prefix. The resulting bound has three contributions: (i) a bridge loss from discarding the window; (ii) a variance-type tail for the prefix; and (iii) a regime-dependent smoothing term (Esseen, bounded density, or cancellation of the third cumulant). Optimizing the window length yields rates that are explicit in the sample size. In the fixed-base (q-adic) case we recover Delange's product and obtain effective convergence bounds; the same scheme applies unchanged to Cantor numeration systems. We also include a brief guide indicating when each regime is preferable.