Return of $k$-bonacci random walks

TL;DR

This paper investigates the return probabilities and fractal dimensions of $k$-bonacci random walks on integers, providing new insights into their recurrence behavior and geometric properties, with specific analysis of tribonacci walks.

Contribution

It introduces a detailed analysis of return probabilities and fractal dimensions for $k$-bonacci based random walks, including specific results for tribonacci sequences.

Findings

Derived the probability of return for $k$-bonacci random walks.

Calculated Hausdorff, packing, and box-counting dimensions of the return sets.

Analyzed the return behavior of tribonacci random walks to the sequence's first term.

Abstract

In this work, the probability of return for random walks on $\mathbb{Z}$, whose increment is given by the $k$-bonacci sequence, is determined. Also, the Hausorff, packing and box-counting dimensions of the set of these walks that return an infinite number of times to the origin are given. As an application, we study the return for tribonacci random walks to the first term of the tribonacci sequence.