The main contribution of this dissertation is two-fold. First, we establish general Cramér type moderate deviation theorems for a class of Studentized non-linear statistics, including Student's t-statistic, Studentized U- and L-statistics as prototypical examples, by developing a novel randomized concentration inequality. As a direct consequence, a sharp moderate deviation for Studentized U-statistics is obtained. Second, we study the asymptotic behaviors of the largest magnitude of the off-diagonal entries of the sample correlation matrix, under the ultra-high dimensional setting. In particular, we obtain necessary and sufficient conditions for the law of large numbers, and also establish limiting distributions under the same sufficient condition. The proofs are based on both...[ Read more ]

The main contribution of this dissertation is two-fold. First, we establish general Cramér type moderate deviation theorems for a class of Studentized non-linear statistics, including Student's t-statistic, Studentized U- and L-statistics as prototypical examples, by developing a novel randomized concentration inequality. As a direct consequence, a sharp moderate deviation for Studentized U-statistics is obtained. Second, we study the asymptotic behaviors of the largest magnitude of the off-diagonal entries of the sample correlation matrix, under the ultra-high dimensional setting. In particular, we obtain necessary and sufficient conditions for the law of large numbers, and also establish limiting distributions under the same sufficient condition. The proofs are based on both classical and self-normalized moderate deviations, the Stein-Chen method and the aforementioned randomized concentration inequality.