Two strong limit theorems for processes with independent increments

Abstract

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), -∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that (ξ(t) - ξ(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 ξ. It addresses the question: Under what conditions does (2T)-1∫-TTX(t)X(t + τ)dt converge as T → ∞ for all τ for almost all sample functions? © 1982.