Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {[xi](t), -[infinity] < t < [infinity]} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that ([xi](t) - [xi](0))/H(t) --> 0 for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is [xi]. It addresses the question: Under what conditions does converge as T --> [infinity] for all [tau] for almost all sample functions?