Inconstancy of finite and infinite sequences

TL;DR

This paper explores the concept of inconstancy as a measure of variability in finite and infinite sequences, revisiting Crofton's theorem and applying it to classical binary sequences to better understand their fluctuation complexity.

Contribution

It introduces the inconstancy measure for sequences, highlighting its relevance over traditional criteria, and applies it to various classical sequences including automatic and Sturmian sequences.

Findings

Inconstancy effectively captures sequence variability.

Crofton's theorem underpins the inconstancy measure.

Classical sequences exhibit diverse inconstancy values.

Abstract

In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy-)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent than other criteria for measuring its variational complexity. We also compute the inconstancy of classical binary sequences including some automatic sequences and Sturmian sequences.