Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation

Abstract

We study the convergence of Bernstein type operators leading to two results. The first: The kernel Kn of the Bernstein–Durrmeyer operator at each point x∈(0,1) — that is Kn(x,t)dt — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions f of bounded variation.