Poisson's theorem

The authors present three methods for proving Poisson's theorem. The first method is based on papers of L. Takács [J. Amer. Statist. Assoc. 62 (1967), 102–113; MR0217832] and J. Galambos [J. Appl. Probab. 11 (1974), 219–222; MR0358923], the second uses results of D. A. Freedman [Ann. Probab. 2 (1974), 256–269; MR0370694] and M. R. Leadbetter [Z. Wahrsch. Verw. Gebiete 28 (1973/74), 298–309; MR0362465], and the third method follows the considerations contained in another paper by Galambos [ibid. 32 (1975), no. 3, 197–207; MR0380941]. The paper contains known theorems but some of the proofs are new.