Implicit Renewal Theory and Power Tails on Trees

: We extend Goldie's (1991) Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to R D = N i=1 CiRi + Q, R D = N i=1 CiRi Q, and similar recursions, where (Q, N, C1, . . . , CN ) is a nonnegative random vector with N {0, 1, 2, 3, . . . } {}, and {Ri}i1 are iid copies of R, independent of (Q, N, C1, . . . , CN ); here denotes the maximum operator. AMS 2000 subject classifications: Primary 60H25; secondary 60J80, 60F10, 60K05. Keywords and phrases: Implicit renewal theory; weighted branching processes; multiplicative cascades; stochastic recursions; power laws; large deviations; stochastic fixed point equations.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Predrag R. Jelenkovic, Mariana Olvera-Cravioto
• Keywords: CORR 2010, Education, Nonnegative Random Vector, Power Tail Asymptotics, Recursions