Summary: It is shown that the class of \(\alpha\)-decomposable distributions of M. Loève [Bull. Soc. Math. Fr. 73, 107–126 (1945; Zbl 0063.03608)] provides all the possible marginal distributions for stationary first-order autoregressive (AR(1)) processes. Integer-valued AR(1) processes are paid special attention. In that context, discrete \(\alpha\)-decomposable distributions are introduced and characterized by limit properties. Several examples are discussed.