Abstract In this paper, we consider limiting properties concerning linear operators of probabilistic type. Specifically, we show that gamma operators are limits of inverse Beta operators and that Bleimann–Butzer–Hahn, Szász and Baskakov operators are limits of generalized Bleimann–Butzer–Hahn operators. By duality, these results are closely related to the convergence in the total variation distance of the probability measures involved. In each case, rates of convergence are given.