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Abstract. We consider two dimensional and three dimensional semi-infinite tubes made of "Lambertian" material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in three dimensions. An observer looking into the three dimensional tube will see "infinitely bright" spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away.

We study solutions to the stochastic fixed point equation X d = AX + B when the coefficients are nonnegative and B is an "inverse exponential decay" (IED) random variable. We provide theorems on the left tail of X which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the IED class. We describe the lower envelope for these ARMA processes.

We consider light ray reflections in d-dimensional semi-infinite tube, for d ≥ 3, made of Lambertian material. The source of light is placed far away from the exit, and the light ray is assumed to reflect so that the distribution of the direction of the reflected light ray has the density proportional to the cosine of the angle with the normal vector. We present new results on the exit distribution from the tube, and generalizations of some theorems from an earlier article, where the dimension was limited to d = 2 and 3.