Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn with n ≧ 1.

The object of this paper is to characterize an information theoretic measure associated with three probability distributions known as ‘Information Improvement’ through a functional equation which arises by considering the additive property of the measure. This measure has been extensively used in economic analysis.

The Lévy distance, L(F,G), between two distribution functions F and G has the important property that convergence of L(Fn,F) is equivalent to convergence in distribution. The fact that L(F,G) is not invariant under a change of scale has been thought to be a disadvantage. However, simple bounds on the Lévy distance between the transformed distribution functions can be found.