Local time, upcrossing time and weak cutpoints of a spatially inhomogeneous random walk on the line

2023/06/26 by Huaming Wang, Wang, Hua-Ming · 1 citation
Decision Sciences · Mathematics · #60G50 #60J10 #60J55 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2306.14376

openalex publication_date 2023/06/26 · openalex created_date 2023/06/28 · openalex updated_date 2026/07/28

Abstract

In this paper, we study a transient spatially inhomogeneous random walk with asymptotically zero drifts on the lattice of the positive half line. We give criteria for the finiteness of the number of points having exactly the same local time and/or upcrossing time and weak cutpoints (a point x is called a weak cutpoint if the walk never returns to x-1 after its first upcrossing from x to x+1). In addition, for the walk with some special local drifts, we also give the order of the expected number of these points in [1,n]. Finally, we show that, when properly scaled, the number of these points in [1,n] converges in distribution to a random variable with the standard exponential distribution. Our results answer three conjectures related to the local time, the upcrossing time, and the weak cutpoints proposed by E. Csáki, A. Földes, P. Révész [J. Theoret. Probab. 23 (2) (2010) 624-638].

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• Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes

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