Output sum of transducers: Limiting distribution and periodic fluctuation

TL;DR

This paper analyzes the asymptotic behavior of sequences generated by transducers, including their expected value, variance, periodic fluctuations, and distribution, with applications to digital sequences and recursions.

Contribution

It provides a general framework for analyzing the asymptotic properties of transducer-defined sequences, including their fluctuations and distribution, extending previous digital sequence analyses.

Findings

Periodic fluctuation is Hölder continuous and often nowhere differentiable.

Sequences are asymptotically normally distributed for many transducers.

Derived a formula for Fourier coefficients of the fluctuation function.

Abstract

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in $[0, N)$. Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework. Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is H\"older continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.