The limit distribution in the \(q\)-CLT for \(q\,\geqslant \,1\) is unique and can not have a compact support (Q2832462)

A generalization of the central limit theorem, called the \(q\)-CLT (for some \(1\leq q<2\)) was developed by the first author et al. [Milan J. Math. 76, 307--328 (2008; Zbl 1182.60011)] for sequences of random variables of the form NEWLINE\[NEWLINE Z_N=\frac{X_1+\dots+X_N}{N^\frac{1}{2(2-q)}}\,, NEWLINE\]NEWLINE where \(X_1,\ldots,X_N\) are identically distributed, and one of a number of different \(q\)-independence properties holds for this sequence. The limiting distribution in the \(q\)-CLT is a generalization of the Gaussian.NEWLINENEWLINEThe present paper fills a gap in the presentation of the \(q\)-CLT by proving that the limiting distribution is unique and cannot have a compact support. Various auxiliary results (for example, concerning the \(q\)-Fourier transform, or the variance and quasivariance of the random variables appearing in the \(q\)-CLT) are also proved.