Tail behavior of solutions of linear recursions on trees

Consider the linear nonhomogeneous fixed-point equation R=D∑i=1NCiRi+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 ZN=∑i=1NCi or Q are regularly varying with index −α<−1 and E[∑i=1NCiα]<1. This work complements previous results showing that P(R>t)∼Ht−α provided there exists a solution α>0 to the equation E[∑i=1N|Ci|α]=1, and both Q and ZN have lighter tails.