Logarithmic tails of sums of products of positive random variables bounded by one

In this paper we show under weak assumptions that for $R1+M_1+M_1{M}_2+\dots$, where $P(M\in [0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim Cx\ln P(M>1-\frac{1}{x})$ as $x\to \mathrm{\infty }$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln P(M>1-\frac{1}{x})$. Random variable $R$ satisfies the stochastic equation $R1+MR$ with $M$ and $R$ independent, thus this result fits into the study of tails of iterated random equations, or more specifically, of perpetuities.