Implicit renewal theorem for trees with general weights

Consider distributional fixed point equations of the form R=Df(Q,Ci,Ri,1≤i≤N), where f(⋅)f(⋅) is a possibly random real-valued function, N∈{0,1,2,3,…}∪{∞}N∈{0,1,2,3,…}∪{∞}, {Ci}i∈N{Ci}i∈N are real-valued random weights and {Ri}i∈N{Ri}i∈N are iid copies of RR, independent of (Q,N,C1,C2,…)(Q,N,C1,C2,…); =D represents equality in distribution. Fixed point equations of this type are important for solving many applied probability problems, ranging from the average case analysis of algorithms to statistical physics. We develop an Implicit Renewal Theorem that enables the characterization of the power tail behavior of the solutions RR to many equations of multiplicative nature that fall into this category. This result extends the prior work in Jelenković and Olvera-Cravioto (2012) [16], which assumed nonnegative weights {Ci}{Ci}, to general real-valued weights. We illustrate the developed theorem by deriving the power tail asymptotics of the solution RR to the linear equation R=D∑i=1NCiRi+Q.