Journal of Mathematical Sciences, Vol. 146, No. 4, 2007 REMARKS ON PROOFS OF THE L ´ EVY–KHINTCHINE FORMULAS FROM THE POINT OF VIEW OF THE GENERALIZED FUNCTIONS Z.(Vladimir) Volkovich 1 UDC 519.2 1. Introduction The L´ evy–Khintchine formula is one of the most common and popular guidelines in probability theory since it provides a decryption for a lot of significant objects such as: the infinitely divisible laws, the infinitesimal operators of semi-groups, etc. Known proofs of this formula employ various concepts, based on the structure of the mentioned distributions, such as: an approximation with the accompanying laws (see, for example, [7]), the shift-compact property of the divisors set (see [11]), the Choquet theorem on the representation by means of the external points of a compact convex set (see [5]). One of the principal contributions of the last method is the use of the notion of “negative-definite functions” that allows a consideration of the infinitely divisible distributions from a similar point of view to the classical Bochner theorem on the integral representation of the positive-definite functions. This approach was discussed earlier in [1]. The study of the possible stochastic centering of the sums of real independent random variables (such as the super- stable distributions of V. M. Zolotarev) leads to an extension of the concepts of an infinitely divisible law. Such a generalization has been proposed by author in [14, 16] as the V -infinitely divisible distributions. Two possible approaches to the study of the representations of the mentioned distribution from the point of view of the theory of generalized functions are discussed in the present paper. The principal point of our consideration is the relation between V -infinitely divisible distributions and the so-called conditionally positive-definite functions. These functions have been introduced in the fundamental paper [9] and have found a widespread purpose in approximation theory. Integral representations of conditionally positive-definite functions can be applied for studying the L´ evy– Khintchine formulas and vice versa. However, in this paper, we focus only on application of the generalized function theory. From our point of view, such an approach allows one to easily understand and interpret these formulas as a unit general object. The paper begins with an explanation of the notation used in this paper and gives a brief description of the V -infinitely divisible distributions. The second part of the paper is devoted to proving and interpretating the general Akhiezer theorem, and the last part consists of a construction of the L´ evy–Khintchine formulas for the V -infinitely divisible distributions from the point of view of generalized function theory. 2. Notation Denote by D 0 the set of all test functions (i.e., infinitely differentiable complex functions having compact support) on the real line R 1 . Let D k , k ≥ 1, be subsets of the functions satisfying, for all 0 ≤ j ≤ k - 1,  x j f (x) dx =0. (1) Definition 1. A generalized function h is called m-positive definite (m-p.d.) if  h(x - y)f (x) f (y) dx dy ≥ 0 (2) for each f ∈ D m . Note that the 0-positive-definite function is a regular positive-definite function, and that if h is 1-positive definite then -h is a so-called, “negative-definite function,” which could be obtained as a logarithm (see [1, 7]) of a characteristic function of an infinitely divisible distribution. As was mentioned above, these functions arise in approximation theory and are called conditionally positive-definite functions. Proceedings of the Seminar on Stability Problems for Stochastic Models, Jurmala, Latvia, 2004, Part I. 6054 1072-3374/07/1464-6054 c  2007 Springer Science+Business Media, Inc.