Precise tail asymptotics of fixed points of the smoothing transform with general weights

We consider solutions of the stochastic equation $R{=}_d{\sum }_{i=1}^N{A}_i{R}_i+B$, where $N>1$ is a fixed constant, $A_i$ are independent, identically distributed random variables and $R_i$ are independent copies of $R$, which are independent both from $A_i$’s and $B$. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being $\mathbb{E}\mathrm{\mid }{A}_1{\mathrm{\mid }}^{\alpha }=1\mathrm{/}N$ and $\mathbb{E}\mathrm{\mid }{A}_1{\mathrm{\mid }}^{\alpha }\log \mathrm{\mid }{A}_1\mathrm{\mid }>0$, the limit ${\lim }_{t\to \mathrm{\infty }}{t}^{\alpha }\mathbb{P}[\mathrm{\mid }R\mathrm{\mid }>t]=K$ exists. In the present paper, we prove positivity of $K$.