Tail behavior of solutions of linear recursions on trees

Consider the linear nonhomogeneous fixed-point equation R = D ∑ i = 1 N C i R i + Q , where ( Q , N , C 1 , C 2 , … ) is a random vector with N ∈ { 0 , 1 , 2 , 3 , … } ∪ { ∞ } , C i ≥ 0 for all i ∈ N , P ( | Q | > 0 ) > 0 , and { R i } i ∈ N is a sequence of i.i.d. random variables independent of ( Q , N , C 1 , C 2 , … ) 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 , C 1 , C 2 , … ) . This paper investigates the settings where either Z N = ∑ i = 1 N C i or Q are regularly varying with index − α < − 1 and E [ ∑ i = 1 N C i α ] < 1 . This work complements previous results showing that P ( R > t ) ∼ H t − α provided there exists a solution α > 0 to the equation E [ ∑ i = 1 N | C i | α ] = 1 , and both Q and Z N have lighter tails.