Bridging a Gap in Independent Component Analysis

Independent Component Analysis (ICA) is a powerful method which aims at representing a given random signal as a sum of independent sources. The engineering community, however, works at the distribution function or characteristic function level while makes assertions at the random variable level. This legitimacy of this jump has never been established and consists of a longstanding gap in the ICA literature. In this paper, it is proved that existence of a factorization of characteristic function does imply existence of a corresponding decomposition of random variable into independent sum and thus the gap is bridged. The proof relies on two nontrivial results from probability theory.