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We introduce asymptotic analogues of the Rademacher and martingale type andcotype of Banach spaces and operators acting on them. Some classical localtheory results related, for example, to the `automatic-type' phenomenon, thetype-cotype duality, or the Maurey-Pisier theorem, are extended to theasymptotic setting. We also investigate operator ideals corresponding to theasymptotic subtype/subcotype. As an application of this theory, we provide asharp version of a result of Brooker and Lancien by showing that any twistedsum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk power type$\max\{p,q\}$.

Operator ideals and three-space properties of asymptotic ideal seminorms