  Based on the Chernoff approximation, we provide a general approximation
result for convex monotone semigroups which are continuous w.r.t. the mixed
topology on suitable spaces of continuous functions. Starting with a family
$(I(t))_{t\geq 0}$ of operators, the semigroup is constructed as the limit
$S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f$ and is uniquely determined by the
time derivative $I'(0)f$ for smooth functions. Moreover, we identity explicit
conditions for the generating family $(I(t))_{t\geq 0}$ that are transferred to
the semigroup $(S(t))_{t\geq 0}$ and can easily be verified in applications. It
turns out that there is a structural connection between Chernoff type
approximations for semigroups and LLN and CLT type results for convex
expectations including large deviation results as well.
