Exact rate of convergence for increments of random fields

The rate of convergence in strong limit theorems for maximal increments of random fields on parallelepipeds of big volume $a_{N}$ ($\lim\frac{a_{N}}{\log{N}}=\infty$, $\lim\frac{\log\frac{N}{a_{N}}}{\log_{2}N}=\infty$) is investigated. We consider random fields with finite moment generating function in right neighborhood of zero.