The maxima of partial sums indexed by squares and rectangles over lattice points and random cubes are studied in this paper. For some of these problems, the dimension (d = 1, d = 2 and d ≥ 3) significantly affects the limit behavior of the maxima. However, for other problems, the maxima behave almost the same as their one-dimensional counterparts. The tools for proving these results are large deviations, the Chen-Stein method, number theory and inequalities of empirical processes.
- Chen-Stein method
- Inequalities of empirical processes
- Large deviations
- Number theory