Noneuclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities

This is an expository paper. The derivation of the ordinary central limit theorem using the Fourier transform on the real line is reviewed. Harmonic analysis on the Poincaré-Lobatchevsky upper half plane H is sketched. The Fourier inversion formula on H reduces to that for the classical integral transform of F. G. Mehler (1881, Math. Ann. 18, 161-194) and V. A. Fock (1943, Compt. Rend. Acad. Sci. URSS Dokl N. S. 39, 253-256), for example. This result is then used to solve the heat equation on H, producing a non-Euclidean analogue of the density function for the Gaussian or normal distribution on H. The non-Euclidean central limit theorem for rotation invariant distributions on H with an application to the statistics of long transmission lines is also discussed.