Asymptotic behavior of increments of random fields

Some results on the asymptotic behavior of increments of a $d$-dimensional random field are proved. Let $N$ and $a_N\in\{1,2,\dots \}$ be fixed and let $S_N^{\star}$ be the maximum increment of a $d$-dimensional random field of independent identically distributed random variables evaluated for $d$-dimensional rectangles $(i,j]=\{k\colon i<k\leq j\}$ such that $\vert j\vert\leq N$ and $\vert j-i\vert=a_N$. Denote also by $S_N$ the maximum increment evaluated for rectangles such that $\vert j-i\vert\leq a_N$.

We determine the asymptotic almost sure behavior of random variables $S_N$ and $S_N^{\star}$. Steinebach (1983) proved a similar result for the case of rectangles belonging to the cube $(0,N^{1/d}]$ (of volume $N$) and under the condition that $a_N=O(N^{\delta})$ as $N\to\infty$ for all $\delta\in(0,1)$. Note that the sequence $S_N$ is monotone in this case.

We also consider the cases where $a_N\sim C\log N$ or $a_N=O(\log N)$.