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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$x \in (0, 1)$ $\unicode{x2013}$ that is $K_n(x, t) dt$ $\unicode{x2013}$ oncestandardised converges to the normal distribution. The second result computesthe pointwise limit of a generalised Bernstein-Durrmeyer operator applied to$\unicode{x2013}$ possibly discontinuous $\unicode{x2013}$ functions $f$ ofbounded variation.

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