Asymptotic theory (statistics)

In statistics, asymptotic theory, or large sample theory, is a generic framework for assessment of properties of estimators and statistical tests. Within this framework it is typically assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as n → ∞.

In practical applications, asymptotic theory is applied by treating the asymptotic results as approximately valid for finite sample sizes as well. Such approach is often criticized for not having any mathematical grounds behind it, yet it is used ubiquitously anyway. The importance of the asymptotic theory is that it often makes possible to carry out the analysis and state many results which cannot be obtained within the standard “finite-sample theory”.

Most statistical problems begin with a dataset of size n. The asymptotic theory proceeds by assuming that it is possible to keep collecting additional data, so that the sample size would grow infinitely:

Under this assumption many results can be obtained that are unavailable for samples of finite sizes. As an example consider the law of large numbers. This law states that for a sequence of iid random variables X₁, X₂, …, the sample averages ${\displaystyle \scriptstyle {\overline {X}}_{n}}$ converge in probability to the population mean E[X_i] as n → ∞. At the same time for finite n it is impossible to claim anything about the distribution of ${\displaystyle \scriptstyle {\overline {X}}_{n}}$ if the distributions of individual X_i’s is unknown.

...
Wikipedia