Second-order behaviour for self-decomposable distributions with two-sided regularly varying densities (Q2135211)

Let \(\mu\) be a self-decomposable probability distribution on the real line whose probability density is regularly varying at \(+\infty\) of order \(-(1+\alpha)\) and regularly varying of order \(-(1+\beta)\) at \(-\infty\), where \(\alpha, \beta \geq 0\). Denote the Lévy measure of \(\mu\) by \(\nu\). The paper provides a second order expansion for the tail \(\nu((x,\infty))\) in terms of the tail \(\mu((x,\infty))\) as \(x\to\infty\), which for most cases of \(\alpha\) and \(\beta\) is of the form \[\nu((x,\infty)) = \mu((x,\infty)) (1 + K_\mu(x) + o(K_\mu(x)))\] as \(x\to\infty\), where \(K_\mu\) is a function that is given explicitly in terms of the tail behaviour of the density of \(\mu\). As a consequence, for a self-decomposable distribution with two-sided regularly varying density, an asymptotic expansion of the tail of \(\mu^{\ast t}\) in terms of the tail of \(\mu\) is given. Here, \(\mu^{\ast t}\) denotes the distinguished \(t\)'th convolution power of \(\mu\).