# Characteristic Functions and Classical Limit Theorems

• Olav Kallenberg
Part of the Probability and Its Applications book series (PIA)

## Abstract

Uniqueness and continuity theorem; Poisson convergence; pos itive and symmetric terms; Lindeberg’s condition; general Gaussian convergence; weak laws of large numbers; domain of Gaussian attraction; vague and weak compactness.

## References

1. The central limit theorem (a name first used by Pölya (1920)) has a long and glorious history, beginning with the work of De Moivre (1733–56), who obtained the now-familiar approximation of binomial probabilities in terms of the normal density function. Laplace (1774, 1812–20) stated the general result in the modern integrated form, but his proof was incomplete, as was the proof of Chebyshev (1867, 1890).Google Scholar
2. The first rigorous proof was given by Liapounov (1901), though under an extra moment condition. Then Lindeberg (1922a) proved his fundamental Theorem 5.12, which in turn led to the basic Proposition 5.9 in a series of papers by Lindeberg (1922b) and Levy (1922a–c). Bernstein (1927) obtained the first extension to higher dimensions. The general problem of normal convergence, regarded for two centuries as the central (indeed the only) theoretical problem in probability, was eventually solved in the form of Theorem 5.15, independently by Feller (1935) and Lévy (1935a). Slowly varying functions were introduced and studied by Karamata (1930).Google Scholar
3. Though characteristic functions have been used in probability theory ever since Laplace (1812–20), their first use in a rigorous proof of a limit theorem had to wait until Liapounov (1901). The first general continuity theorem was established by Levy (1922c), who assumed the characteristic functions to converge uniformly in some neighborhood of the origin. The definitive version in Theorem 5.22 is due to Bochner (1933). Our direct approach to Theorem 5.3 may be new, in avoiding the relatively deep Helly selection theorem (1911–12). The basic Corollary 5.5 was noted by Cramer and Wold (1936).Google Scholar
4. Introductions to characteristic functions and classical limit theorems may be found in many textbooks, notably Loève (1977). Feller (1971) is a rich source of further information on Laplace transforms, characteristic functions, and classical limit theorems. For more detailed or advanced results on characteristic functions, see Lukacs (1970).Google Scholar