The converse bounded-cell conjecture for strongly β-mixing max-stable random fields

Let $\eta$ be a stationary max-stable random field, and let $C(0)$ denote its cell containing the origin. The field is strongly $\beta$-mixing if for every compact set $S\subset\mathcal{X}$, $\lim_{r\to+\infty}\beta_r(S)=0$, where $\beta_r(S)$ is the $\beta$-mixing coefficient between $S$ and the part of the field at distance at least $r$ from $S$. Converse bounded-cell conjecture. If $\eta$ is a strongly $\beta$-mixing stationary max-stable random field, then $C(0)$ is almost surely bounded. The preceding proposition establishes the forward implication from almost-sure boundedness of $C(0)$ to strong $\beta$-mixing; this conjecture asserts the converse, whose status is not resolved in the supplied text.