对于$q \ge 1$，CLT中的极限分布是唯一的，不能有紧支集
The limit distribution in the $q$-CLT for $q \ge 1$ is unique and can not have a compact support
摘要：
在Umarov、Tsallis和Steinberg（2008）的论文中，介绍了傅立叶变换的一种推广，称为$q$ -傅立叶变换，并将其用于证明$q$ -广义中心极限定理（$q$ -CLT）。随后，Hilhorst（2009年和2010年）说明了$q>1$的$q$ -傅立叶变换在密度函数空间中是不可逆的。事实上，使用不变性原则，他构建了一个家庭的密度与相同的$q$ -傅立叶变换，并指出，“因此，$q$ -中心极限定理达不到其既定目标”。在那里构造的分布具有紧支撑。我们证明了$q$ -CLT中的极限分布是唯一的，并且不可能有紧支集。这个结果排除了所有可能的反例，可以使用不变性原理构造，并填补了Hilhorst提到的空白。
英文摘要：
In a paper by Umarov, Tsallis and Steinberg (2008), a generalization of the Fourier transform, called the $q$-Fourier transform, was introduced and applied for the proof of a $q$-generalized central limit theorem ($q$-CLT). Subsequently, Hilhorst illustrated (2009 and 2010) that the $q$-Fourier transform for $q>1$ is not invertible in the space of density functions. Indeed, using an invariance principle, he constructed a family of densities with the same $q$-Fourier transform and noted that "as a consequence, the $q$-central limit theorem falls short of achieving its stated goal". The distributions constructed there have compact support. We prove now that the limit distribution in the $q$-CLT is unique and can not have a compact support. This result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by Hilhorst.
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