A Braching Random Walk with a Barrier

Abstract

Suppose that a child is likely to be weaker than its parent, and child who is too weak will not reproduce. What is the condition for a family to survive? Let b denote the mean number of children a viable parent will have; we suppose that this is independent of strength of strength as long as strength is positive. Let F denote the distribution of the change in strength from parent to child, and define h = supq (- log U eqt dF(t))). We show that the situation is black or white: 1) If b < eh then P(family line dies) = 1, 2) If b > eh then P(family survives) > 0. Define f(x) := E(number of members in the family | initial strength x). We show that if b < eh, then there exists a positive constant C such that limx ƀ e-ax f(x) = C where a is the smaller of the (at most) two positive roots of b U est dF(t) = 1. We also find an explicit expression for f(x) when the walk is on a lattice and is skip-free to the left. This process arose in an analysis of rollback-based simulation, and these results are the foundation of that analysis.