1. Introduction

Along with the prevalence of probabilistic methods in approximation theory the study of approximation of operators by other operator sequences, or the study of limiting properties of operator sequences, has attracted more and more attention. It was shown by Stancu [15] in the 1960s that Szász operators and Baskakov operators are limiting operators of Stancu–Mühlbach operator sequences in an appropriate sense. Later, Butzer and Hahn [4, 5, 11] established rates of convergence of some probabilistic type operators towards their limiting operators as applications of their general approximation theorems in probability theory. In 1988, Khan [13] proved that Szász operators are limiting operators of Bleimann–Butzer–Hahn operator sequences. Recently de la Cal, Luquin, Adell and San Miguel investigated the limiting properties of a series of probabilistic type operator sequences and exhibited that their limiting operators might be Bernstein, Szász, Baskakov, gamma or beta operators [1, 2, 6, 7, 8].