[Paper Review] The origin of infinitely divisible distributions: from de Finetti's problem to Levy-Khintchine formula

Francesco Mainardi, Sergei Rogosin|ArXiv.org|Jan 12, 2008

Complex Systems and Time Series AnalysisEconomics, Econometrics and Finance81 references21 citations

TL;DR

This paper traces the historical development of infinitely divisible distributions from de Finetti's foundational 1929 work to the canonical Lévy-Khintchine formula, establishing the mathematical lineage of characteristic functions for stable and infinitely divisible laws. It provides the first English translation of Khintchine's seminal 1937 paper, which formalized the integral representation of the logarithm of a characteristic function, thereby completing the theoretical framework for infinitely divisible distributions.

ABSTRACT

The article provides an historical survey of the early contributions on infinitely divisible distributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms developed in the thirties by Kolmogorov, Levy and Khintchine. Particular attention is paid to single out the personal contributions of the above authors that were published in Italian, French or Russian during the period 1929-1938. In Appendix we report the translation from the Russian into English of a fundamental paper by Khintchine published in Moscow in 1937.

Motivation & Objective

• To reconstruct the historical evolution of infinitely divisible distributions from de Finetti's 1929 work to the Lévy-Khintchine formula.
• To clarify the individual contributions of de Finetti, Kolmogorov, Lévy, and Khintchine during 1929–1938, particularly in Italian, French, and Russian publications.
• To provide the first English translation of Khintchine's 1937 paper, a foundational work in the derivation of the Lévy-Khintchine formula.
• To establish the mathematical equivalence between the canonical form of the logarithm of a characteristic function and the infinite divisibility of the corresponding distribution.
• To support future research by offering a detailed, annotated bibliography and biographical context for the key pioneers in the field.

Proposed method

• Conducts a historical survey of early works on infinitely divisible distributions, focusing on original publications by de Finetti, Kolmogorov, Lévy, and Khintchine.
• Analyzes the evolution of the concept of infinite divisibility, starting from de Finetti’s problem of decomposing distributions into sums of i.i.d. random variables.
• Derives the Lévy-Khintchine formula by showing that the logarithm of a characteristic function can be expressed as an integral involving a Lévy measure, a Gaussian component, and a drift term.
• Uses approximation techniques with truncated integrals (e.g., ∫|u|>ε) to handle singularities and prove convergence to the canonical form.
• Applies the limit process as ε → 0 to show that the expression log φ(t) = itγ − at² + ∫(e^{itu}−1−itu/(1+u²)) dG(u) is the logarithm of a characteristic function.
• Validates the sufficiency of the Lévy-Khintchine representation by proving that any such function corresponds to an infinitely divisible distribution.

Experimental results

Research questions

• RQ1 How did the concept of infinite divisibility emerge from de Finetti’s 1929 work, and what was its original formulation?
• RQ2 What was the role of Kolmogorov and Lévy in advancing the theory of infinitely divisible distributions following de Finetti’s initial results?
• RQ3 How did Khintchine’s 1937 paper contribute to the formal derivation of the Lévy-Khintchine formula, and why was this result pivotal?
• RQ4 Why is the Lévy-Khintchine formula considered canonical, and how does it unify the representation of all infinitely divisible distributions?
• RQ5 What is the significance of the English translation of Khintchine’s 1937 paper for modern researchers in probability theory and stochastic processes?

Key findings

• The term 'infinitely divisible' first appeared in print in a 1936 unpublished thesis by G.M. Bawly, though it was not consistently applied.
• Khintchine provided the first formal definition of an infinitely divisible distribution as one that can be represented as the sum of n i.i.d. random variables for any positive integer n.
• The Lévy-Khintchine formula was established as the canonical representation of the logarithm of the characteristic function of an infinitely divisible distribution.
• The formula takes the form log φ(t) = itγ − at² + ∫(e^{itu}−1−itu/(1+u²)) dG(u), where G is a bounded, nondecreasing function.
• The proof demonstrates that any function of this form is the logarithm of a characteristic function, thereby confirming the infinite divisibility of the corresponding distribution.
• The limit process as ε → 0 is uniformly controlled, ensuring convergence and validity of the canonical form even at u = 0, with the help of a supplementary estimate by Gnedenko on the uniformity of the integral tail.