Limit Theorems for Convex Expectations

BasedBlessing, J.   Kupper, M. on the Chernoff approximation, we provide a general approximation result for convex monotone semigroups. Starting with a family \((I(t))_{t\ge 0}\) of operators, the semigroup is constructed as the limit \(S(t)f:=\lim _{n\rightarrow \infty }I(\frac{t}{n})^n f\) and is uniquely determined by the time derivative \(I'(0)f\) for smooth functions. We identify explicit conditions for the generating family \((I(t))_{t\ge 0}\) that are transferred to the semigroup \((S(t))_{t\ge 0}\) and can easily be verified in applications. Furthermore, there is a structural link between Chernoff type approximations for nonlinear semigroups and law of large numbers type results for convex expectations.