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 $d \geq 3$) 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.
"Maxima of partial sums indexed by geometrical structures." Ann. Probab. 30 (4) 1854 - 1892, October 2002. https://doi.org/10.1214/aop/1039548374