A unified theory is constructed which describes the a.s. (almost surely) behavior of increments of stochastically continuous homogeneous processes with independent increments. This theory includes the strong law of large numbers, the Erdös-Rényi law, the Shepp law, the Csörgo- Révész law, and the law of the iterated logarithm. The range of applicability of the results is extended from several particular cases to the whole class of stochastically continuous homogeneous processes with independent increments.