The law of large numbers for free identically distributed random variables
成果类型：
Article
署名作者：
Bercovici, H; Pata, V
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1996
页码：
453-465
关键词：
摘要：
Let X(1),X(2),... be a sequence of free identically distributed random variables, with common distribution mu. It was shown by Lindsay and Pata, in a more general context, that a sufficient condition for the weak law of large numbers to hold for the sequence X(1), X(2),... is that (t-->infinity)lim t mu({x: \x\ > t}) = 0. We show that this condition is necessary as well as sufficient. Even though the condition is identical with the corresponding one for commuting independent variables, the proof of the result uses the analytical techniques of free convolution theory, and it is quite different from the proof of the commutative theorem due to Kolmogorov [cf. Feller (1971).