Affine random equations and the stable \(\left( {1 \over 2} \right)\) distribution (Q2754983)

The authors study the affine random equation \(X=S+L^2X\) in distribution for a given \(\mathbb R\)-valued random variable \(X\) with a stable \((1/2)\) distribution and for unknown \(\mathbb R_+\)-valued random variables \(S\) and \(L\). The cases where \(S\) and \(L\) are independent or infinitely divisible are of particular interest. The main result of this paper states that \(E(L)\leq 1\) is necessary and sufficient for the existence of \(S\) on some probability space.