Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.
Asymptotic theory is concerned with the properties of estimators and test statistics in large samples which are assumed to tend towards infinity in size. This allows to obtain complicated estimators and tests which would not be available in small samples. Note that asymptotic theory is only an approximation with small samples, and it is not always a good approximation. Examples of asymptotic properties of estimators are consistency, regularity or their asymptotic distribution. Frequently used concepts in asymptotic theory include the weak and strong law of large numbers, the central limit theorem, certain classes of expansions (e.g. Taylor, Edgeworth, von Mises) and the Delta method.