Limit distributions of norms of vectors of positive i. i. d. random variables
Let \((X_n)_{n\in\mathbb N}\) be an i.i.d.~sequence of positive random variables, \(c\) a positive constant and \((p(n))_{n\in\mathbb N}\) a sequence in \((0,\infty]\). The present paper is concerned with convergence of the suitably centered and normalized \(c p(n)\)-norm of the vector \((X_1, \ldots, X_n)\) to a limit distribution \(F_c\). The limit distributions \(F_c\) are determined in the case that \(X_1\) is in the maximum domain of attraction of the Fréchet distribution, where \(p(n)=1\), and in the case that \(X_1\) is in the maximum domain of attraction of the Weibull distribution, where \((p(n))_{n\in\mathbb N}\) is a sequence depending on the upper tail behaviour of \(X_1\), but not on \(c\). Centering and normalization constants are given explicitly. It is shown that in both cases, \(F_c\) converges in distribution to the standard normal distribution as \(c \to 0+\), and to the Fréchet or Weibull distribution, respectively, as \(c \to \infty\). For sequences in the maximum domain of attraction of the Gumbel distribution, the example of exponential random variables is considered, and conjectures for the general case are given.
independent identically distributed positive random variables
scientific article; zbMATH DE number 1906003
scientific article