Chains of saturated models in AECs

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: $\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{r}\mathbf{e}\mathbf{m}$ If $K$ is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $\lambda$:

• The union of an increasing chain of $\lambda$-saturated models is $\lambda$-saturated.
• There exists a type-full good $\lambda$-frame with underlying class the saturated models of size $\lambda$.
• There exists a unique limit model of size $\lambda$. Our proofs use independence calculus and a generalization of averages to this non first-order context.