# Anomalous recurrence properties of many-dimensional zero-drift random walks

@article{Georgiou2016AnomalousRP, title={Anomalous recurrence properties of many-dimensional zero-drift random walks}, author={Nicholas Georgiou and Mikhail Menshikov and Aleksandar Mijatovi{\'c} and Andrew R. Wade}, journal={Advances in Applied Probability}, year={2016}, volume={48}, pages={99 - 118} }

Abstract Famously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so…

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## References

SHOWING 1-10 OF 28 REFERENCES

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

- Mathematics
- 2012

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for…

The oscillating random walk

- Mathematics, Physics
- 1974

{Yn;n=0, 1, ...} denotes a stationary Markov chain taking values in Rd. As long as the process stays on the same side of a fixed hyperplane E0, it behaves as an ordinary random walk with jump measure…

Excursions and path functionals for stochastic processes with asymptotically zero drifts

- Mathematics
- 2013

We study discrete-time stochastic processes (Xt) on [0,∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about…

Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains

- Mathematics, Physics
- 2008

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary.…

On recurrence and transience of self-interacting random walks

- Mathematics
- 2013

Let µ1,...,µk be d-dimensional probabilitymeasures in ℝd with mean 0. At each time we choose one of the measures based on the history of the process and take a step according to that measure. We give…

Foundations of modern probability

- Mathematics
- 1997

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit…

The Problem of the Random Walk

- Nature
- 1905

THIS problem, proposed by Prof. Karl Pearson in the current number of NATURE, is the same as that of the composition of n iso-periodic vibrations of unit amplitude and of phases distributed at…

A generalized Pearson random walk allowing for bias

- Mathematics
- 1974

We calculate asymptotic values of the first two moments of a planar walk in which the step lengths depend on the direction of motion. The model is suggested by experiments on the locomotion of…

Foundations of modern probability (2nd edn), by Olav Kallenberg. Pp. 638. £49 (hbk). 2002. ISBN 0 387 95313 2 (Springer-Verlag).

- PhilosophyThe Mathematical Gazette
- 2004

distribution, queuing theory, random walks, and so on. On many topical issues he is prepared to admit that there are no definitive answers, considering, inter alia, the following questions: how…

Criteria for the recurrence or transience of stochastic process. I

- Mathematics
- 1960

Conditions for recurrence or transience of Markov chains are studied. Criteria of intermediate generality are established for some classes of processes with known transition probabilities which arise…