Investigated is asymptotic almost surely behavior of increments (AASBI) of stochastically continuous homogeneous processes with independent increments (SCHPwII). The universal formula is derived for normalizing function in the limit theorems on AASBI SCHPwII. It allows constructing the integral theory of AASBI SCHPwII including certain considered statistical laws and thus enables to expand AASBI description results for the whole class of SCHPwII.