On asymptotic B-convexity and infratype

In this note, we introduce and study the notions of asymptotic B-convexity and asymptotic infratype $p$, and we prove asymptotic analogs of a series of results due to Giesy \cite{Giesy66} and Pisier \cite{Pisier74}. In particular, we give a simplified proof of an asymptotic version of Pisier's $\ell_1$-theorem that was originally proven by Causey, Draga, and Kochanek in \cite{CDK19}. We also briefly discuss the notion of asymptotic stable type.