## 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.

### MSC:

 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
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### References:

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