Stable central limit theorem in total variation distance
Let \(X_1,X_2,\ldots\) be independent and identically distributed \(d\)-dimensional random vectors in the domain of normal attraction of a stable distribution with \(\alpha\in(0,2)\), and whose distribution is locally lower bounded by the Lebesgue measure. Under some boundedness or symmetry assumptions on the associated spectral measure, the authors establish rates of convergence of the distribution of (an appropriate normalisation of) \(X_1+\cdots+X_n\) to a stable law in total variation distance. The case in which the \(X_i\) have a symmetric Pareto distribution is used to show optimality of these convergence rates, and in particular that in the case \(\alpha=1\) the optimal rate is \(n^{-1}\).
scientific article; zbMATH DE number 7957627
scientific article