The Borel-Cantelli is an important theorem about sequences of events and is commonly used in convergence theorems in probability theory. The original Borel-Cantelli consists of two parts, which were proposed by Émile Borel and Francesco Paolo Cantelli respectively. In the SW project, we will introduce several different extensions of the Borel-Cantelli Lemma including our original results. In addition, we categorize the generalizations of the second Borel-Cantelli Lemma into two groups and summarize the important expansions of the union of some probabilities that are often used in the proof of the generalizations of the Borel-Cantelli Lemma. We also present our new, simple proof of the growth rate of partial maxima of uniform random variables. In 1961, Barndorff-Nielsen first gave the characterization criterion for characterizing the growth rate of partial maxima. We extend Barndorff-Nielsen’s characterization criterion and get a slightly stronger result. In the end, we propose some open problems and introduce our future work in this field.
Our original work in the SW project includes proposing several new, strong generalizations of the Borel-Cantelli Lemma, categorizing the different generalizations of the second Borel-Cantelli Lemma into two groups properly, and presenting a new, simple proof of a more general characterization criterion for lower class sequences for partial maxima. |
