Tail behavior of solutions of linear recursions on trees

Consider the linear nonhomogeneous fixed-point equation R D = N � i=1 Ci Ri + Q, where (Q, N , C1, C2, . . .) is a random vector with N ∈ {0, 1, 2, 3, . . .} ∪ {∞}, Ci ≥ 0 for all i ∈ N, P(|Q| > 0) > 0, and {Ri }i∈N is a sequence of i.i.d. random variables independent of (Q, N , C1, C2, . . .) having the same distribution as R. It is known that R will have a heavy-tailed distribution under several different sets of assumptions on the vector (Q, N , C1, C2, . . .). This paper investigates the settings where either Z N = �N i=1 Ci or Q are regularly varying with index −α < −1 and E ��N i=1 C α i � < 1. This work complements previous results showing that P(R > t) ∼ Ht−α provided there exists a solution α > 0 to the equation E ��N i=1 |Ci |α � = 1, and both Q and Z N have lighter tails. c � 2012 Elsevier B.V. All rights reserved. MSC: 60H25; 60J80; 60F10; 60K05