Kac’s central limit theorem by Stein’s method

In 1946, Mark Kac proved a Central Limit type theorem for a sequence of random variables that were not independent. The random variables under consideration were obtained from the angle-doubling map. The idea behind Kac’s proof was to show that although the random variables under consideration were not independent, they were what he calls statistically independent (in modern terminology, this concept is called long range independence). Using that observation, Kac showed that the sample averages of the random variables, suitably normalized, converges to the standard normal distribution. In this paper, we give a new proof of Kac’s result by applying Stein’s method. We show that the normalized sample averages converge to the standard normal distribution in the Wasserstein metric, which in particular implies convergence in distribution.