Tails of Lévy measure of geometric stable random variables (Q1294772)

This paper deals with the so-called geometric stable distributions in \(R^1\), i.e. with infinitely divisible distributions \(GS(\alpha,\beta)\) whose characteristic function has the form \(\psi_{\alpha,\beta}(t)=(1-\log \varphi_{\alpha,\beta}(t))^{-1}\), where \(\varphi_{\alpha,\beta}(t)\) is a characteristic function of some stable distribution with parameters \(\alpha\in (0,2]\), \(|\beta|\leq 1\). Let \(\Lambda=\Lambda_{\alpha,\beta}\) be the Lévy measure of \(GS(\alpha,\beta)\). Denote by \(\Lambda_+=\Lambda((u,\infty))\) and \(\Lambda_-=\Lambda((-\infty,-u))\) its right and left tails, resp. The asymptotic behaviour around zero of \(\Lambda_+\) and \(\Lambda_-\) is investigated. The precise expression for \(\lim_{u\to 0} \Lambda_{\pm}/\ln u\) is obtained for various values of \(\alpha\) and \(\beta\). For instance, if \(\alpha \not =1\), \(\beta=0\), then \(\lim_{u\to 0} \Lambda_{\pm}/\ln u=-( \alpha/2)\). These facts imply, in particular, the exponential rate of convergence in series expansion of geometric stable Lévy processes.