Globally coupled chaos violates the law of large numbers but not the central-limit theorem

The title statement is numerically shown for a globally coupled chaotic system. With an increasing number of elements, N, the distribution of the mean field approaches a Gaussian distribution, but the decrease of its mean-square deviation with N stops for large N. This violation of the law of large numbers is found to be caused by the emergence of a subtle coherence among elements, as is measured by the mutual information. With the inclusion of noise, the law of large numbers is restored. The mean-square deviation decreases in proportion to ${\mathit{N}}^{\mathrm{-}\mathrm{\beta }}$ with an exponent β<1 depending on the noise strength.