Lecture 01 : the Central Limit Theorem

Central and Local Limit Theories in Markov Dependent Random Variables

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

Lecture 4 : Law of Large Number and Central Limit Theorem ( LaTeX prepared by Jing

1 Probability Bounds for PF and PM In the previous lectures we have studied various detection methods. Starting from this lecture, we want to take a step further to analyze the performance of these detection methods. In order to motivate ourselves to learn a set of new tools called Large Deviation Theory, let us first review some “standard” tools, namely the Law of Large Number and the Central ...

Lecture 13 : Empirical Central Limit Theorems

This lecture will complete our coverage of empirical process theory. We will state without proof a more abstract version of the Donsker Theorem, that is covered in detail in (Pollard, 1984, §7.5), and supply two examples of its application. 1 Setup Recall the entropy integral from last lecture, where we developed the Chaining Lemma for checking the conditions for stochastic equicontinuity.