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

Let { A i } 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 Z = d ∑ i = 1 N A i Z i , where Z , Z 1 , Z 2 , … are independent and identically distributed random variables, independent of N , A 1 , A 2 , … . We assume E [ ∑ i = 1 N A i ] = 1 and E [ ∑ i = 1 N A i log A i ] = 0 (the boundary case), then it is known that all nonzero solutions have infinite mean. We obtain new results concerning behavior of their tails.