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

Burdzy, Krzysztof; Tadić, Tvrtko

J. Theor. Probab. 31, No. 1, 466-493 (2018).

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.

Reviewer: Anatoliy Swishchuk (Calgary)

MSC:

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

Keywords:

random reflections; stopped random walks; tail of the arccosine distribution; overshoot

Citations:

Zbl 1373.60080

References:

[1]	Angel, O; Burdzy, K; Sheffield, S, Deterministic approximations of random reflectors, Trans. Am. Math. Soc., 365, 6367-6383, (2013) · Zbl 1408.37063
[2]	Arnold, BC; Groeneveld, RA, Some properties of the arcsine distribution, J. Am. Stat. Assoc., 75, 173-175, (1980) · Zbl 0427.62008
[3]	Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1987) · Zbl 0617.26001
[4]	Burdzy, K., Tadić, T.: Can one make a laser out of cardboard? Ann. Appl. Probab. Arxiv:1507.00961 (2016) · Zbl 1186.37049
[5]	Comets, F; Popov, S; Schütz, GM; Vachkovskaia, M, Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal., 191, 497-537, (2009) · Zbl 1186.37049
[6]	Comets, F; Popov, S; Schütz, GM; Vachkovskaia, M, Knudsen gas in a finite random tube: transport diffusion and first passage properties, J. Stat. Phys., 140, 948-984, (2010) · Zbl 1197.82058
[7]	Comets, F; Popov, S; Schütz, GM; Vachkovskaia, M, Quenched invariance principle for the Knudsen stochastic billiard in a random tube, Ann. Probab., 38, 1019-1061, (2010) · Zbl 1200.60091
[8]	Doney, RA, Moments of ladder heights in random walks, J. Appl. Probab., 17, 248-252, (1980) · Zbl 0424.60072
[9]	Evans, SN, Stochastic billiards on general tables, Ann. Appl. Probab., 11, 419-437, (2001) · Zbl 1015.60058
[10]	Folland, G.B.: Real analysis. Pure and Applied Mathematics (New York), 2nd edn. Wiley, New York (1999). Modern techniques and their applications, A Wiley-Interscience Publication · Zbl 0924.28001
[11]	Veraverbeke, N, Asymptotic behaviour of Wiener-Hopf factors of a random walk, Stoch. Process. Appl., 5, 27-37, (1977) · Zbl 0353.60073