Asymptotic properties of symmetric stable distributions with small index (Q1917615)

Let \(\xi\) have the symmetric stable distribution with characteristic function \(\varphi (t)= \exp (-|t|^\alpha)\), \(0< \alpha\leq 2\). As is known, if \(\alpha\to 0\), then \(P\{|\xi|^{- \alpha}< x\}\to 1-\exp (-x)\), \(x>0\). Let \(g(x)= g(x, \alpha)\) denote the distribution density of \(|\xi|^{- \alpha}\). The estimation of \(g(x, \alpha)\) as \(\alpha\to 0\) in the domain of large deviations of an arbitrary order (i.e. for \(x= x(\alpha)\to \infty\)) is analyzed. The list of references may be extended by [the reviewer, Theory Probab. Appl. 30, No. 3, 643 (1985)] in which some problems in case \(\alpha \uparrow 2\) were analyzed.