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Department of Mathematical Sciences

Seminar Archives

Statistics Seminars: Zero-drift random walks with anomalous recurrence properties

Presented by Nicholas Georgiou, Durham University

12 May 2014 02:00 in CM221

The symmetric simple random walk (SRW) in Z^d is known to be recurrent for dimension d = 1
or 2, and transient for all larger dimensions. A related random walk, the
Pearson-Rayleigh random walk, is defined on R^d for d at least 2 and proceeds by taking
unit length steps, but in a random direction chosen uniformly from the continuum of
directions available (compare with the SRW where there are a finite set of 2d directions).
As with the SRW, this walk is recurrent in dimension 2, but transient for dimension 3 or
greater.

Both these walks have zero drift, and it is natural to ask whether this property
determines the recurrence/transience for general (non-homogeneous) random walks. We show
that in dimension 2 or more this is not the case: we describe a family of non-homogeneous
random walks having zero-drift (themselves being generalisations of the Pearson-Rayleigh
random walk) that for each dimension includes walks that are recurrent and walks that are
transient. In particular, there are random walks with the following anomalous asymptotic
behaviour:

* zero-drift random walks in dimension 2 that are transient,
* zero-drift random walks in dimension 3 or greater that are recurrent.

We analyse the walk by representing the location as a radial part together with a
projection onto the unit sphere. We determine the recurrence/transience of the walk by
studying the radial process, and using the scaling limit of the walk we prove an ergodic
theorem for the projected process on the unit sphere. We make use of the skew-product
decomposition of the scaling limit, reminiscent of the skew-product decomposition of
Brownian motion.