Chapter IX - General properties. Characteristic functions

Summary

1. For a distribution in a one-dimensional space, the only possible discontinuities arise from discrete points which, in terms of the mechanical interpretation used in Chapter II, are bearers of positive quantities of mass. As soon as the number of dimensions exceeds unity, the question of the discontinuities becomes, however, more complicated. Thus in a k-dimensional space, the whole mass may be concentrated to a sub-space of less than k dimensions (line, surface, …), though there is no single point that carries a positive quantity of mass.

Given a random variable X = (ξ₁ …,ξ_k) in the k-dimensional space R_k, we denote as in Chapter II the corresponding pr.f. by P (S) and the d.f. by F (x₁ …, x_k). Just as in the case k = 1, there can at most be a finite number of points A such that P (A) > a > 0, and hence at most an enumerable set of points B such that P (B) > 0. We shall call this set the point spectrum of the distribution.

According to II, § 3, every component ξ_i of X is itself a random variable, and the corresponding (one-dimensional) distribution is found by projecting the original distribution on the axis of ξ_i.