On tails of fixed points of the smoothing transform in the boundary case

Let {Ai} be a sequence of random positive numbers, such that only N first of them are strictly positive, where N is a finite a.s. random number. In this paper we investigate nonnegative solutions of the distributional equation , where Z,Z1,Z2,... are independent and identically distributed random variables, independent of N,A1,A2,.... We assume and (the boundary case), then it is known that all nonzero solutions have infinite mean. We obtain new results concerning behavior of their tails.