Gaussian Convergence and Inequalities
This chapter describes a simple Gaussian limit theory; namely we restate simple central limit theorems together with applications and moment/exponential inequalities for partial sums behaving asymptotically as Gaussian random variables. A relevant reference for the whole chapter is Petrov (Limit theorems of probability theory. Sequences of independent random variables, Oxford University Press, Oxford, 1975), results without a precise reference should be found in this reference, and the others are in Hall and Heyde (Martingale limit theory and its application, Academic Press, London, 1980). Topics related to empirical processes are covered by van der Vaart and Wellner (Weak convergence and empirical processes, Springer, New York, 1998) and Rosenblatt (Stochastic curve estimation. NSF-CBMS regional conference series in probability and statistics, 1991).