of observations with certain desirable asymptotic properties. An effective formalization of probability theory was given in a 1933 monograph of Kolmogorov that was based on measure theory.1 An attempt to get a rigorous format for the von Mises collective led in the 1960s to an approach using algorithmic information theory by Kolmogorov and others.
Feller’s research made use of the measure theoretic foundations of probability theory as did most mathematical work. Feller’s first published paper in probability theory (1936) obtained necessary and sufficient conditions for the central limit theorem of probability theory. This was a culmination of earlier work of DeMoivre, Laplace, and Liapounov. The concern is with the asymptotic behavior of the partial sums Sn= X1+…+Xn of a sequence of independent random variables X1,X2,…,Xn,…. Assuming that means mk=EXk and variances exist, Feller showed that the normalized and centered sums (with the summands (Xj-mj)/Bn uniformly asymptotically negligible) in distribution converge to the normal law
as n→∞ if and only if the Lindeberg condition
(Fk is the distribution function of Xk) as n→∞ is satisfied for each ε>0.
The actual fluctuating behavior of the sequence Sn was first properly formulated by Borel in 1909. After initial advances by Hausdorff, Hardy and Littlewood, and Khinchin, Kolmogorov obtained a law of the iterated logarithm, which