Stable-law and logarithmic central limit conjecture for symmetric observables

Let $\gamma\in(\frac12,1)$, let $v:X\to\mathbb{R}^d$ and $h:X\to G$ be the observable and cocycle from the system, and let $\operatorname{Fix} h(0)=\{w\in\mathbb{R}^d:h(0)w=w\}$. Write $\phi_n$ for the associated Birkhoff-sum process. Assume that $v(0)\notin(\operatorname{Fix} h(0))^\perp$. Stable-law and logarithmic central limit conjecture. If $\gamma\in(\frac12,1)$, then $n^{-\gamma}\phi_n$ converges in distribution to a $d$-dimensional stable law of order $\alpha=1/\gamma$. If $\gamma=\frac12$ and $v(0)\notin(\operatorname{Fix} h(0))^\perp$, then $(n\log n)^{-\frac12}\phi_n$ converges in distribution to a $d$-dimensional normal distribution. These predictions describe the anomalous-diffusion regimes excluded by the preceding central limit theorem, and are motivated by heuristic arguments; the source provides no proof or resolution.