A few ways to destroy entropic chaoticity on Kac's sphere (Q2017232)

\textit{M. Kac} [in: Proc. 3rd Berkeley Sympos. Math. Statist. Probability 3, 171--197 (1956; Zbl 0072.42802)] called ``\(f\)-chaotic'' any sequence \((F_N)_{N\in\mathbb{N}}\) of symmetric (i.e., invariant under permutations of arguments) probability densities on the sphere \(\mathbb{S}^{N-1}(\sqrt{N})\), such that for any \(k\in\mathbb{N}^*\) the marginal of \(F_N\) evaluated at \((v_1,\dots, v_k)\in\mathbb{R}^k\) goes weakly to \(f(v_1)\times\cdots\times f(v_k)\) as \(N\to\infty\). Denote by \(\gamma\) the standard Gaussian kernel on \(\mathbb{R}\), and by \(H(f,\gamma)\) the relative entropy of the probability density \(f\), and more generally by \(H(\mu,\sigma)\) the relative entropy of two measures \(\mu, \sigma\). Then, \(F_N\) as above is said to be ``entropically chaotic'' if \[ \lim_{N\to\infty} {1\over N} H(F_N\mid\sigma_N)= H(f,\gamma), \] where \(\sigma_N\) denotes the uniform probability measure on \(\mathbb{S}^{N-1}(\sqrt{N})\). Focussing mainly on those \(F_N\) that have the product form \(c_N \prod^N_{j=1} f(v_j)\), for \(f\) making them \(f\)-chaotic, the author exhibits three examples of \(f\)-chaotic but not entropically chaotic sequences \((F_N)\).