A study of the Hajek-Renyi inequality and its applications.

摘要

Inequalities are at the heart of mathematical and statistical theory. No inequality is completely perfect, but the Hajek-Renyi inequality, which is the main subject of this thesis, is arguably the closest to absolute perfection of all the inequalities within all theories of probability. It has many applications in proving limit theorems, and examples of these are presented in this thesis. The strong law of large numbers for sequences of random variables and the strong growth rate for sums of random variables were obtained through utilizing the Hajek-Renyi inequality. This thesis will further extend and improve the proof of the strong law of large numbers. Additionally, the approach, utilizing the Hajek-Renyi inequality to prove limit theorems, is also applied to the weak law of large numbers for tail series.