Geodesic
On the local growth of random fields with independent increments
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 184-191

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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. 
@article{TVP_1979_24_1_a18,
     author = {N. M. Zin\v{c}enko},
     title = {On the local growth of random fields with independent increments},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {184--191},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_1_a18/}
}
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N. M. Zinčenko. On the local growth of random fields with independent increments. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 184-191. http://geodesic.mathdoc.fr/item/TVP_1979_24_1_a18/