Regular variation in the branching random walk

TL;DR

This paper investigates the tail behavior of the normalized moment generating functions in a supercritical branching random walk, establishing regular variation properties under mild conditions without using Laplace transforms.

Contribution

It proves that the tail distribution of various functionals of the branching random walk regularly varies with the same exponent, using non-analytic methods.

Findings

Tail distributions of functionals are regularly varying with the same exponent.

Results hold under mild moment restrictions and regular variation assumptions.

The proofs do not rely on Laplace-Stieltjes transforms.

Abstract

Let $\{\mm_n, n=0,1,...\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\mm_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, $\su |W-W_n|$, $\su |W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that $\rP\{W_1>x\}$ regularly varies at $\infty$ it is proved that $\rP\{AW_n>x\}$ regularly varies at $\infty$ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace-Stieltjes transforms. The result on the tail behaviour of $W$ is established in two distinct ways.