Tail behavior of laws stable by random weighted mean

Abstract

Let ( N , A 1 , A 2 , … ) be a sequence of random variables with N ∈ N ∪ { ∞ } and A i ∈ R + . We are interested in asymptotic properties of solutions of the distributional equation Z = ∑ i = 1 N A i Z i , where Z i are nonnegative random variables independent of each other and independent of ( N , A 1 , A 2 , … ) , each has the same distribution as Z which is unknown. For a solution Z ⩾ 0 with finite mean, we show that under a natural moment condition, the regular variation of P ( Z > x ) ( x → ∞ ) is equivalent to that of P ( Y 1 > x ) , where Y 1 = ∑ i = 1 N A i . The results generalize the corresponding theorems of Bingham and Doney (1974, 1975) [1,2] and de Meyer (1982) [6] on Galton–Watson processes and Crump–Mode–Jirina processes, and improve those of Iksanov and Polotskiy (2006) [7] on branching random walks.