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

We study the convergence of Bernstein type operators leading to two results. The first: The kernel $K_n$ of the Bernstein-Durrmeyer operator at each point $x\in (0,1)$ $\text{unicode}x2013$ that is $K_n(x,t)dt$ $\text{unicode}x2013$ once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein-Durrmeyer operator applied to $\text{unicode}x2013$ possibly discontinuous $\text{unicode}x2013$ functions $f$ of bounded variation.