  We have obtained some upper bounds for the probability distribution of
extremes of a self-similar Gaussian random field with stationary rectangular
increments that are defined on the compact spaces. The probability
distributions of extremes for the normalized self-similar Gaussian random
fields with stationary rectangular increments defined in ${\mathbb{R}}^2_+$
have been presented. In our work we have used the techniques developed for the
self-similar fields and based on the classical series analysis of the maximal
probability bounding from below for the Gaussian fields.
