On asymptotic behaviour of increments of random fields

We derive universal strong laws for increments of sums of i.i.d. random variables with multidimensional indices where an exponential moment does not exist. Our theorems yield the strong law of large numbers, the law of the iterated logarithm and the Csörgő-Révész laws for random fields. New results are obtained for distributions from domains of attraction of a normal law and completely asymmetric stable laws with index $\alpha\in(1,2)$.