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January 1996 The law of large numbers for free identically distributed random variables
Hari Bercovici, Vittorino Pata
Ann. Probab. 24(1): 453-465 (January 1996). DOI: 10.1214/aop/1042644726


Let $ X_1, X_2,\ldots$ 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 $X_1, X_2,\ldots$ is that

$$\lim_{t\to \infty}t\mu(\{x: |x|>t\}) = 0>$$

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.


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Hari Bercovici. Vittorino Pata. "The law of large numbers for free identically distributed random variables." Ann. Probab. 24 (1) 453 - 465, January 1996.


Published: January 1996
First available in Project Euclid: 15 January 2003

zbMATH: 0862.46036
MathSciNet: MR1387645
Digital Object Identifier: 10.1214/aop/1042644726

Primary: 46LJ50 , 60L05
Secondary: 47C15

Keywords: distribution , Free convolution , free random variables , Law of Large Numbers

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 1 • January 1996
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