Random reflections in a high-dimensional tube. (English) Zbl 1391.60098

The paper is a continuation of the authors paper in [Ann. Appl. Probab. 27, No. 4, 1951–1991 (2017; Zbl 1373.60080)], where Lambertian reflections in a semi-inifinte tube were introduced and studied, and the focus was on the two-dimensional case. The main results were concerned with the properties of the light ray at the exit time.
The present paper is devoted to the \(d\)-dimensional case for any \(d>3\). The authors show that the exit distributions are similar in the quantitative sense for all \(d\geq 3\). Some new results are also derived, including overshoot of random walks, arccosine distribution and products of random variables having this distributions, and distributions with regular varying tails.


60G50 Sums of independent random variables; random walks
60K05 Renewal theory
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37H99 Random dynamical systems


Zbl 1373.60080
Full Text: DOI arXiv


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