On lower tail probabilities of positive random sums

Let $(\xi_k)_{k\geq1}$ be an i.i.d. sequence of positive random variables with $O$-varying distribution function at 0. Further let $(a_k)_{k\geq1}$ be a sequence of positive weights such that the positive random sum $S=\sum_{k=1}^\infty a_k\xi_k$ exists almost surely. Without assuming finite second moments, the author determines the asymptotic behaviour of the left tail $P\{S<\varepsilon\}$ as $\varepsilon\downarrow0$ and of the density function at 0 in terms of the asymptotic behaviour of the Laplace transform at $\infty$ using Escher transforms. It turns out that necessarily $S$ belongs to the Type I domain of attraction for minima. An application is given for $\xi_k=\eta_k^2$ with $\alpha$-stable random variables $\eta_k$ on the real line