Representation of Probability Distributions as the Convolution of Symmetric and Completely Asymmetric Parts

摘要

Let F, G, and H be probability distributions on the line each having finite variance and suppose G is symmetric. F is completely asymmetric (c.as.) if the equation F = G*H implies G = delta sub 0, i.e. is degenerate. It's proven that F can always be written F = G*H where H is c.as., but this representation may not be unique. Examples of singular and absolutely continuous (with respect to Lebesgue measure) c.as. distributions are given. Some extensions of these ideas are mentioned. (Author)