Abstract In this paper we characterize random variables which are stable but not strictly in the sense of generalized convolution. We generalize results obtained Jarczyk and Misiewicz (J Theoret Probab 22:482-505, 2009), Mazurkiewicz 18:837-852, 2005), Oleszkiewicz (in Milman VD Schechtman Lecture Notes Math. 1807, Geometric Aspects Functional Analysis (2003), Israel Seminar 2001–2002, Springer...