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

In this paper, we show under weak assumptions that for $R\mathop = \limits^d 1 + {M_1} + {M_1}{M_2} + \cdots $, where P(M . [0, 1]) = 1 and Mi are independent copies of M, we have ln.(R > x) ~ Cx ln.(M > 1 - 1/x) as x . .. The constant C is given explicitly and its value depends on the rate of convergence of ln.(M > 1 - 1/x). Random variable R satisfies the stochastic equation $R\mathop = \limits^d 1 + MR$ with M and R independent, thus this result fits into the study of tails of iterated random equations, or more specifically, perpetuities.