ABSTRACT
This chapter begins with a review of the theoretical framework of Book II for the construction of a probability space on which a countable collection of independent, identically distributed random variables can be defined, and thus on which limit theorems of various types can be addressed. The first section then addresses weak convergence of various distribution function sequences. Among those studied are the Student T, Poisson, DeMoivre-Laplace, and a first version of the central limit theorem, as well as Smirnov’s result on uniform order statistics, a general result on exponential order statistics, and finally a limit theorem on quantiles. The next section generalizes the study of laws of large numbers of Book II using moment defined limits, and proves a limit theorem on extreme value theory identified in that book. The final section studies empirical distribution functions, and in particular derives the Glivenko-Cantelli theorem on convergence of such empirical distributions to the underlying distribution function. Kolmogorov’s theorem on the limiting distribution of the maximum error in an empirical distribution is also discussed, as are related results.
