Limit theorems for elephant random walks remembering the very recent past, with applications to the Takagi-van der Waerden class functions

TL;DR

This paper links probabilistic models of elephant random walks with the oscillatory and differentiability properties of Takagi-van der Waerden functions, providing new insights into their behavior through probabilistic analysis.

Contribution

It introduces new limit theorems for elephant random walks with recent past memory and applies these to analyze the properties of Takagi-van der Waerden functions.

Findings

Precise estimates for oscillations of Takagi-van der Waerden functions.

Necessary and sufficient conditions for localization of ERWVRP.

Complete characterization of differentiability properties of the functions.

Abstract

We study the Takagi-van der Waerden functions $f_r (x)$, a well-known class of continuous but nowhere differentiable functions, from probabilistic point of view. As an application of elephant random walks remembering the very recent past (ERWVRP, a.k.a. symmetric correlated random walks), we obtain precise estimates for the oscillations of $f_r (x)$. We also establish a result on the necessary and sufficient condition for localization of the ERWVRP with variable step length, which can be applied to obtain a complete description of the differentiability properties of the Takagi-van der Waerden class functions.