On the local growth of random fields with independent increments

The paper deals with the behaviour of a random field $\xi (t,s)$ with independent increments in the neighbourhood of zero. The classes of upper and lower functions for such fields are defined. It is proved that the real function $\varphi (t,s)$ under some additional assumptions is upper (lower) function if the integral
$$ \int_0^{t_0}\int_0^{s_0} [ts]^{-1} \mathbf P\{\xi(t,s)>\varphi(t,s)\}\,ds\,dt $$
is convergent (divergent). As a consequence we obtain the integral criterion for the 2-parameter Brownian motion and the law of iterated logarithm for this field. All results are generalized for the case off $n$-dimensional parameter.