On some random walk problems

2018/02/19 by Chak Hei Lo, Lo, Chak Hei
Mathematics · #60F05 #60F15 (Secondary) #60G50 (Primary) 60J05 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1802.06623

openalex publication_date 2018/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In the first part of this thesis, we study a Markov chain on ℝ₊ × S, where ℝ₊ is the non-negative real numbers and S is a finite set, in which when the ℝ₊-coordinate is large, the S-coordinate of the process is approximately Markov with stationary distribution π_i on S. Denoting by μ_i(x) the mean drift of the ℝ₊-coordinate of the process at (x,i) ∈ ℝ₊ × S, we give an exhaustive recurrence classification in the case where ∑_i π_i μ_i (x) → 0, which is the critical regime for the recurrence-transience phase transition. If μ_i(x) → 0 for all i, it is natural to study the Lamperti case where μ_i(x) = O(1/x); in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If μ_i (x) → d_i for d_i ≠ 0 for at least some i, then it is natural to study the generalized Lamperti case where μ_i (x) = d_i + O (1/x). By exploiting a transformation which maps the generalized Lamperti case to the Lamperti case, we obtain a recurrence classification and an existence of moments result for the former. In the second part of the thesis, for a random walk S_n on ℝ^d we study the asymptotic behaviour of the associated centre of mass process G_n = n^-1 ∑_i=1ⁿ S_i. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, G_n is recurrent if d=1 and transient if d ≥ 2. In the transient case we show that G_n has diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which G_n is transient in d=1.

Citations

• Foundations of Modern Probability
• Correlated random walk
• The oscillating random walk
• A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators
• Central Limit Theorem for the self-repelling random walk with directed edges

Related

• Random walks with occasionally modified transition probabilities
• Persistent random walks. II. Functional Scaling Limits
• Recurrent random walks on ℤ with infinite variance: transition probabilities of them killed on a finite set
• Fluctuation theory for Markov random walks
• Recurrence of Markov chain traces