Infinite divisible and stable distributions modulo 1

Introduction. Infinite divisibility and stability in the customary sense is extensively discussed in the literature (see e.g. Lukacs (1970), Feller (1971), Petrov (1975)). Schatte (1983) studies infinite divisibility modulo 2p (mod 2p). He gives a representation theorem for infinite divisible (infdiv) (mod 2p) Fourier-Stieltjes Sequences (FSS's) and a limit theorem for sequences of infdiv (mod 2p) FSS's. Furthermore, under an infinite smallness (mod 2p) condition he considers convergence of sums to infdiv (mod 2p) distributions. In this paper we consider distributions modulo 1 (mod 1). In Section 2 we give some notations, definitions, and properties of FSS's, and in Section 3 we reformulate Schatte' s results for infdiv (mod 1) distributions. From Schatte's representation we deduce in Section 4 two other representations: one similar to the Lévy-Khinchine canonical form, and the other to the Kolmogorov canonical form. In addition, we give a new characterization of infdiv (mod 1) distributions. In Section 5 we define stable (mod 1) distributions and characterize these distributions in two theorems. Finally, in Section 6, we generalize a limit theorem proved by Schatte.