General theorems on rates of convergence in distribution of random variables I. General limit theorems

A note on stochastic methods in connection with approximation theorems for positive linear operators

“…The proof follows from Theorem 3 in [2] similarly as did the previous proof from Theorem 4 in [2]. Taking into account that apparently all known operators which allow probabilistic interpretations arise from identically distributed r.v., let us formulate Theorems 1 and 2 under this restriction, choosing furthermore φ{n) = n~u.…”

“…If one would try to apply Theorem 4 in [2] for r > 2, say r = 3, in order to try to deduce higher order results, this would lead to the assumption…”

“…Nevertheless one can weaken assumption (3.1) to obtain the following THEOREM 2. Let β e (1,2] be such that…”

“…X t results in a positive linear operator L n , they actually arise from more general theorems of probability established in P. L. Butzer-L. Hahn [2,3]. If one looks at these papers, it is easy to see that convergence towards the limiting r.v.…”

“…X/: = X t -E(X t ), leading to E{Xl) = S(X 0 ) = 0, and deduce by Theorem 4 in [2] (case r = 2) that for feC b (R) there exists a constant M > 0 such that…”