Rate of convergence of increments for random fields

The purpose of this paper is to obtain exact convergence rate in the limit theorems for maximal increments of random fields
\begin{align} S_{N,a_{N}}&=\max\Bigl\{\sum _{i<k\leq j}X_{k}:|j|\leq N,|j-i|\leq a_{N}\Bigr\},\notag\\ S^{\star}_{N,a_{N}}&=\max\Bigl\{\sum _{i<k\leq j}X_{k}:|j|\leq N,| j-i|=a_{N}\Bigr\},\notag \end{align}
where $N\in\mathbb{N}$ and $a_{N}=c\log N+\lambda\log_{2} N+o(\log_{2} N)$, $c>c_{0}$, $\lambda\in\mathbb{R}$, $X_{n}$ is a sequence of multi-dimension indexed i.i.d. centered random variables having a finite moment generating function in right neighborhood of zero, $|n|$ is defined by multiplying of coordinates.