| In this dissertation, we first derive some maximal-type probability inequalities for discrete time multidimensionally indexed martingales, both forward and reverse. In particular, we extend a maximal inequality of Chow which gives extensions of well-known results as special cases, including an extension of Doob's inequality. Also, our Chow-type inequality is used to obtain a strong law of large numbers in general form, as well as a Kolmogorov strong law.;In addition to multidimensionally indexed martingales, we study two classes of U-statistics. The first is the well-known class of multi-sample or generalized U-statistics. For this class we apply some of the maximal inequalities mentioned above to establish asymptotic convergence results. The second class is the family of U-statistics based on multidimensionally indexed arrays of random variables. Here, the U-statistic is defined in the usual way except that the random variables on which it is based are multidimensionally indexed. Statistics such as the sample mean or the sample variance of a random field are U-statistics of this kind. Various results are given for this class, including a strong law of large numbers, a law of the iterated logarithm, a central limit theorem and an invariance principle. These results are obtained by exploiting the martingale structure of U-statistics and by making use of our maximal inequalities. |
