arXiv subjects

Yuzaburo Nakano

Publications and source records attributed to Yuzaburo Nakano.

3 recordsLinked to original sources

arXiv2026

An almost sure invariance principle for the Takagi-van der Waerden class functions

The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic point of view. We prove that the local modulus of continuity of $f_{r,a}(x)$ is described by a standard Brownian motion under some regularity assumptions on the weights, as an application of a strong approximation for elephant random walks remembering the very recent past with variable step length.

math.PR↗

arXiv2025

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

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.

math.PR↗

arXiv2025

Elephant random walk with polynomially decaying steps

In this paper, we introduce a variation of the elephant random walk whose steps are polynomially decaying. At each time $k$, the walker's step size is $k^{-γ}$ with $γ>0$. We investigate effects of the step size exponent $γ$ and the memory parameter $α\in [-1,1]$ on the long-time behavior of the walker. For fixed $α$, it admits phase transition from divergence to convergence (localization) at $γ_{c}(α)=\max \{α,1/2\}$. This means that large enough memory effect can shift the critical point for localization. Moreover, we obtain quantitative limit theorems which provide a detailed picture of the long-time behavior of the walker.

math.PR↗