On the empirical measure of the Fourier coefficients with infinite variance data (Q1181087)

Let \(X,X_ 1,\dots,X_ n\) be i.i.d. random variables which are in the domain of attraction of a stable law with index \(\alpha\in(0,2)\). The author studies the limiting behavior of the empirical measure \(P^*_ n\) of a sequence of random variables obtained by Fourier-like transformations of \(X_ 1,\dots,X_ n\). When \(X\) has finite variance, \(P^*_ n\) converges weakly in probability to a non-random normal distribution. When \(X\) has infinite variance, \(P^*_ n\) converges in distribution to a random probability measure provided that \(X\) is in the normal domain of attraction of a stable law, as shown by \textit{D. Freedman} and \textit{D. Lane} [Ann. Stat. 8, 1244-1251 (1980; Zbl 0441.62042)]. The author extends the result of Freedman and Lane to include the case where \(X\) is in the domain of attraction of a stable law.