Remarks on convergence of Morley sequences

We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on equivalences of convergent Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and give a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [P18] on the model-theoretic meaning of Grothendieck's double limit theorem.Comment: (23 pages) In this new version (1) a model-theoretic argument of Claim 2 in Theorem 2.11 is suggested by Predrag Tanovi\'c, (2) Claim 0 in Theorem 2.11 has been proven, (3) Fact 2.10 has been rewritten. (Theorem 4.10 reinforces the main result of the following paper on the model-theoretic meaning of Grothendieck's double limit theorem: arXiv:1703.04761

Generic Stability and Modes of Convergence

We study generically stable types/measures in both classical and continuous logics, and their connection with randomization and modes of convergence of types/measures.Comment: Preliminary version: Comments are welcome