We prove that the norm of a $d$-dimensional L\'evy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\infty)$. We further prove that under the standard $\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the L\'evy process has a finite $(2+\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for L\'evy processes cannot hold without finiteness of $(2+\delta)$-moments for some $\delta>0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.

翻译：我们证明，当且仅当经过适当尺度调整后的$d$维Lévy过程在时刻$t$与标准高斯向量的凸距离关于尺度不变测度$t^{-1} dt$在$[1,\infty)$上可积时，该Lévy过程的范数具有有限二阶矩。进一步证明，在标准$\sqrt{t}$尺度调整下，相应的凸距离可积当且仅当该Lévy过程的范数具有有限$(2+\log)$阶矩。这两个等价关系对多元Kolmogorov距离关于$t^{-1} dt$的可积性同样成立。我们的结果意味着：(I) 对于Lévy过程中心极限定理中凸距离收敛速率的多项式Berry-Esseen界，若不对某个$\delta>0$要求$(2+\delta)$阶矩有限性则不可能成立；(II) 在非正态吸引域中，无论采用何种尺度函数，凸距离关于$t^{-1} dt$的可积性都不可能发生。