Summary: Let \(\{X(t),\;t \geq 0\}\) be a stochastic process, continuous in probability, homogeneous and with independent increments. We consider conditions under which the identical distribution, to within a shift, of two stochastic integrals defined (in the sense of convergence in probability) with respect to the process implies that the process is semi-stable. Our discussion is mostly based on M. Riedel [Adv. Appl. Probab. 12, 689-709 (1980; Zbl 0442.62015)], which concerns itself with stable processes in the same context after identifying some important consequences of the equidistribution assumption. Our main result is Theorem 1. We also examine two particular cases, each in the light of Theorem 1 and independently through more ‘elementary’ arguments. The first (Theorem 2) is a stronger version of a result due to B. L. S. Prakasa Rao and the author [Aequationes Math. 26, 113-119 (1983; Zbl 0538.60021) and Sankhyā, Ser. A 46, 326-338 (1984; Zbl 0569.39002)]and incidentally shows that the technical conditions imposed in (the preamble to) Theorem 1, while sufficient, are not necessary. The second (Theorem 3) is a strong version of a result due to E. Lukacs [J. Appl. Probab. 6, 409-418 (1969; Zbl 0186.500)] on stable processes.