Convergence of Random Variables
Ery Arias-Castro · Cambridge University Press eBooks · 2022
An empirical average will converge, in some sense, to the corresponding expectation. This famous result, called the Law of Large Numbers, can be anticipated based on the concentration inequalities introduced in the previous chapter, but some appropriate notions of convergence for random variables need to be defined in order to make a rigorous statement. Beyond mere convergence, the fluctuations of an empirical average around the associated expectation can be characterized by the Central Limit Theorem, and are known to be Gaussian in some asymptotic sense. The chapter also discusses the limit of extremes such as the maximum of a sample.