The logarithmic correlation-decay threshold for Poisson approximation of largest gaps

Let $f$ be a smooth stationary Gaussian process, let $K(r)$ denote its correlation function, and let the Poisson approximation refer to the approximation for the locations and sizes of its largest gaps. There exists a $\gamma_0>0$ such that the Poisson approximation is valid for every smooth SGP satisfying

$$K(r)(\log r)^{\gamma'}\to 0$$

as $r\to\infty$ for some $\gamma'>\gamma_0$, but not necessarily if

$$K(r)(\log r)^{\gamma'}\to\infty$$

as $r\to\infty$ for some $\gamma'<\gamma_0$. Logarithmic correlation-decay threshold conjecture. There exists a threshold exponent $\gamma_0>0$ separating a regime in which the Poisson approximation holds for all such smooth stationary Gaussian processes from a regime in which it need not hold. The paper establishes the Poisson approximation under stronger decay assumptions, while the precise threshold and the possible failure below it remain open.