Abstract
Let (X n ,n≥1) be a real-valued ergodic stationary stochastic process, and let (Y n =X 1 +…+X n ,n≥1) be the associated random walk. We prove the following: if the sequence of distributions of the random variables Y n /n,n≥1, is uniformly tight (or, more generally, does not have the zero measure as a vague limit point), then there exists a real number c such that the random walk (Y n −nc,n≥1) is recurrent. If this sequence of distributions converges to a probability measure ρ on ℝ (or, more generally, has a nonzero limit ρ in the vague topology), then (Y n −nc,n≥1) is recurrent for ρ−a.e.cℝ.
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Received: 24 September 2001 / Revised version: 1 August 2002 / Published online: 24 October 2002
The first author was partially supported by the FWF research project P14379-MAT.
Mathematics Subject Classification (2000): 37A20, 37A50, 60G10, 60G50
Key words or phrases: Recurrent stationary random walks – Recurrent cocycles
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Greschonig, G., Schmidt, K. Growth and recurrence of stationary random walks. Probab Theory Relat Fields 125, 266–270 (2003). https://doi.org/10.1007/s004400200237
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DOI: https://doi.org/10.1007/s004400200237
Keywords
- Real Number
- Stochastic Process
- Probability Measure
- Random Walk
- Limit Point