Independent Increments and Infinite Divisibility
Regularity and integral representation; Levy processes and subordinators; stable processes and first-passage times; infinitely divisible distributions; characteristics and convergence criteria; approximation of Lévy processes and random walks; limit theorems for null arrays; convergence of extremes
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- Until the 1920s, Brownian motion and the Poisson process were essentially the only known processes with independent increments. In (1924, 1925) Levy introduced the stable distributions and noted that they too could be associated with suitable “decomposable” processes. De Finetti (1929) saw the general connection between processes with independent increments and infinitely divisible distributions and posed the problem of characterizing the latter. A partial solution for distributions with a finite second moment was found by Kolmogorov (1932).Google Scholar
- The complete solution was obtained in a revolutionary paper by Lévy (1934–35), where the “decomposable” processes are analyzed by a virtuosic blend of analytic and probabilistic methods, leading to an explicit description in terms of a jump and a diffusion component. As a byproduct, Levy obtained the general representation for the associated characteristic functions. His analysis was so complete that only improvements in detail have since been possible. In particular, Itô (1942b) showed how the jump component can be expressed in terms of Poisson integrals. Analytic derivations of the representation formula for the characteristic function were later given by Levy (1954) himself, by Feller (1937), and by Khinchin (1937).Google Scholar
- The scope of the classical central limit problem was broadened by Lévy (1925) to a general study of suitably normalized partial sums, obtained from a single sequence of independent random variables. To include the case of the classical Poisson approximation, Kolmogorov proposed a further extension to general triangular arrays, subject to the sole condition of uniformly asymptotically negligible elements. In this context, Feller (1937) and Khinchin (1937) proved independently that the limiting distributions are infinitely divisible. It remained to characterize the convergence to specific limits, a problem that had already been solved in the Gaussian case by Feller (1935) and Levy (1935a). The ultimate solution was obtained independently by Doeblin (1939) and Gnedenko (1939), and a comprehensive exposition of the theory was published by Gnedenko and Kolmogorov (1968).Google Scholar
- The basic convergence Theorem 15.17 for Levy processes and the associated approximation result for random walks in Corollary 15.20 are essentially due to Skorohod (1957), though with rather different statements and proofs. Lemma 15.22 appears in Doeblin (1939a). Our approach to the basic representation theorem is a modernized version of Levy’s proof, with simplifications resulting from the use of basic point process and martingale methods.Google Scholar
- Detailed accounts of the basic limit theory for null arrays are provided by many authors, including Loève (1977) and Feller (1971). The positive case is treated in Kallenberg (1986). A modern introduction to Levy processes is given by Bertoin (1996). General independent increment processes and associated limit theorems are treated in Jacod and Shiryaev (1987). Extreme value theory is surveyed by Leadbetter et al. (1983).Google Scholar