Abstract
A history of the strong law of large numbers certainly began with E. Borel. But BoreFs result was motivated or influenced by Bertrand, Poincare, Wiman, and the others. The main results for this law had ended almost with A. Kolmogorov about 1933. The study of this law consists of two aspects of the development. The one aspect was to deepen the study of the relations between measure theory and the theory of denumerable probabilities. The other is as follows:
Let p be a point of the interval(0,1)and let p=P1P2P3...... be its binary expansion. Let
Xn(p)=｛+1 if pk=1
｛-1 if pk=0
Then Sn=X1+X2+ ……+Xn is the excess frequency of occurrence of the digit 1 among the first n places in the expansion of p. Borel and Cantelli assert almost everywhere Sn = O(n). The enumeration of sharper results indicated the historic development of the problem.
In this paper, these two aspects are described historically. For the theorem of the strong law of large numbers is not a mere theorem, but the processes of its studies are just a history of the probability theory in the early 20th century.
