Showing papers on "Infinite divisibility published in 1974"

On the tails of infinitely divisible distributions

Abstract: Similar theorems are proved in [4] and [2]. Ruegg supposes T(x) to be of the form T(x)=O(exp{--axb}) or T(x)=O(exp{--ax(logx)b}). In his proofs he makes extensive use of the theory of entire functions. Horn considers T(x) satisfying T(x) = O (exp { x M(x)}), where M (x) is non-negative, continuous and ultimately increasing. His proofs depend on straightforward but rather intricate inequalities for the two-sided Laplace transform. In our proof, it seems, more efficient use is made of the infinite divisibility condition. Ruegg's theorems 1 and 2, in fact, hold for all d.f.'s having non-vanishing entire characteristic function (c. f.'s). In Section 3 we prove that an i.d.d.f, with x 2 log T(x)~ ~ , is degenerate by proving that i~ts variance must be zero, as in the case of i.d.d.f. 's on a finite interval (cf. [1], p. 117). In Section 4 we give a very simple proof of a variant of Theorem 1 for distributions on the non-negative integers.

Two Types of Continuity

Abstract: In this paper I am going to deal with two very different kinds of continuity. One is of mathematical kind and it is familiar to every student of calculus; the other was named by Poincare — not very appropriately, as we shall see — physical continuity (le continu physique). While the obvious contrast between these two different types of continuity is fairly well known, its deeper philosophical significance is rarely analyzed. This lack of interest in it is not accidental; it is due to the persistent influence of the intellectual tradition generated by the three centuries of classical science (1600–1900). We shall see that a more subtle epistemological approach together with the emergence of some new and quite unexpected problems in contemporary physics requires another fresh look at the contrast between both types of continuity and the way it was interpreted both by classical science and classical philosophy.

Infinite divisibility and stability of finite semi-Markov matrices

Abstract: In this paper the elements of a semi-Markov matrix A may have support anywhere on the real line, and A(+ co) may be a sub-Markov matrix. The sub-characteristic matrix a of A is the matrix of Fourier-Stieltjes transforms corresponding to A. If A(+ co) = a(O) is a Markov matrix, A is called a matrix distribution funotion and at a characteristic matrix. Infinite divisibility and stability of semi-Markov and sub-characteristic matrices is defined as for distribution functions and characteristic functions.