# On a property of the simple random walk on $\mathbb{Z}$

## TL;DR

This paper investigates the probability that a simple random walk on integers is on the positive side given it has spent a certain percentage of time there, revealing a decomposition of the arcsine law into the Marchenko-Pastur law.

## Contribution

It provides a simple, explicit answer to the probability question and decomposes the arcsine law into the Marchenko-Pastur law using walk classifications.

## Key findings

- The distribution of the time spent on the positive side follows the arcsine law.
- The decomposition links the arcsine law to the Marchenko-Pastur law.
- A new perspective on the structure of random walks on integers.

## Abstract

The subject of this paper is the simple random walk on $\mathbb{Z}$. We give a very simple answer to the following problem: under the condition that a random walk has already spent $\alpha$-percent of the traveling time on the positive side $\mathbb{Z}_{\ge 0}$, what is the probability that the random walk is now on the positive side? The symmetric random walks which step $2n$-times can be decomposed in the following two ways: (1) how many times the walk steps on the positive side, (2) whether the last step is on the positive side or on the negative side. To answer the problem above, we clarify the number of the walks classified by (1) and (2). It has been already known that the distribution of the number indicated by (1) makes the arcsine law. Combining with the decomposition with respect to (2), we obtain a decomposition of the arcsine law into the Marchenko-Pastur law.