Tail behavior of solutions of linear recursions on trees

TL;DR

This paper studies the tail behavior of solutions to a linear recursive distributional equation on trees, focusing on cases where either the sum of coefficients or the additive term are regularly varying, extending previous heavy-tail results.

Contribution

It analyzes the tail behavior of solutions when either the sum of coefficients or the additive term is regularly varying with index -alpha, under specific expectation conditions, complementing prior heavy-tail results.

Findings

Identifies conditions under which the solution R has heavy tails.

Extends previous results to cases with regularly varying components.

Provides asymptotic tail estimates for solutions of the recursion.

Abstract

Consider the linear nonhomogeneous fixed point equation R =_d sum_{i=1}^N C_i R_i + Q, where (Q,N,C_1,...,C_N) is a random vector with N in{0,1,2,3,...}U{infty}, {C_i}_{i=1}^N >= 0, P(|Q|>0) > 0, and {R_i}_{i=1}^N is a sequence of i.i.d. random variables independent of (Q,N,C_1,...,C_N) 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_N). This paper investigates the settings where either Z_N = sum_{i=1}^N C_i or Q are regularly varying with index -alpha < -1 and E[sum_{i=1}^N C_i^alpha] < 1. This work complements previous results showing that P(R>t) Ht^{-alpha} provided there exists a solution alpha > 0 to the equation E[sum_{i=1}^N|C_i|^alpha] = 1, and both Q and Z_N have lighter tails.