The asymptotic behavior of rarely visited edges of the simple random walk

2023/10/25 by Hu, Ze-Chun, Peng, Xue, Song, Renming +1
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2310.16657

Abstract

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on ℤ. Let α(n) be the number of rarely visited edges up to time n. First, we evaluate 𝔼(α(n)), show that n→ 𝔼(α(n)) is non-decreasing in n and that lim_n→+∞𝔼(α(n))=2. Then we study the asymptotic behavior of ℙ (α(n)>a(log n)²) for any a>0 and use it to show that there exists a constant C∈(0,+∞) such that \limsup_n→+∞(α(n))/((log n)²)=C almost surely.

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