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

Let {Ai}{Ai} be a sequence of random positive numbers, such that only NN first of them are strictly positive, where NN is a finite a.s. random number. In this paper we investigate nonnegative solutions of the distributional equation Z=d∑i=1NAiZi, where Z,Z1,Z2,…Z,Z1,Z2,… are independent and identically distributed random variables, independent of N,A1,A2,…N,A1,A2,…. We assume E[∑i=1NAi]=1 and E[∑i=1NAilogAi]=0 (the boundary case), then it is known that all nonzero solutions have infinite mean. We obtain new results concerning behavior of their tails.