Tail behavior, modes and other characteristics of stable distribution (Q1979090)

A family of stable distributions is considered, i.e. distributions with the characteristic functions \(E\exp(i\theta X)= \exp(-|\theta|^\alpha[1+ i\beta\tan(\alpha\pi/2)\text{sign}(\theta)|\theta|^{1-\alpha}-1] \) if \(\alpha\not=1\) and \(E\exp(i\theta X)= \exp(-|\theta|[1+ i\beta(2/\pi)\text{sign}(\theta)\ln|\theta|]) \) if \(\alpha=1\). Denote the corresponding density by \(f(x;\alpha,\beta)\). It is well known that \(f(x;\alpha,\beta)\sim f_{\text{Pareto}}(x;\alpha,\beta)= \alpha(1+\beta)C_\alpha x^{-(1+\alpha)}\) as \(x\to\infty\). The authors analyze numerically the rate of convergence of \(L_{\alpha,\beta}(x)=f(x;\alpha,\beta)/ f_{\text{Pareto}}(x;\alpha,\beta)\to 1\) and analogous characteristics. The plots presented in the article demonstrate quite unusual behavior of these characteristics with abrupt changes. The authors show how this specific behavior of the tails implies the incorrectness of the Hill estimator for the Pareto index \(\alpha\) of a stable distribution if \(\alpha\approx 2\). The behavior of the distribution modes is also investigated numerically.