The Escape Rate of Favorite Edges of Simple Random Walk

2023/03/23 by Hao, Chen-Xu
#60J10 #60J55 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2303.13210

Abstract

Consider a simple symmetric random walk on the integer lattice ℤ. Let E(n) denote a favorite edge of the random walk at time n. In this paper, we study the escape rate of E(n), and show that almost surely \liminf_n→∞\frac|E(n)|√(n)⋅(log n)^-γ equals 0 if γ≤ 1, and is infinity otherwise. We also obtain a law of the iterated logarithm for E(n).

Related

• The asymptotic behavior of rarely visited edges of the simple random walk
• Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk
• Rate of escape of the conditioned two-dimensional simple random walk
• On a maximum of nearest-neighbor random walk with asymptotically zero drift on lattice of positive half line
• How Far Might We Walk at Random?