The law of large numbers for free identically distributed random variables

Abstract

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.