Population size of critical Galton-Watson processes under small deviations and infinite variance

We study the evolution of the population size distribution of a critical Galton-Watson process with infinite variance of the offspring size of particles assuming that the population size is unusually small at the distant moment $n$ of observation.

💡 Research Summary

The paper investigates a critical Galton‑Watson branching process whose offspring distribution has infinite variance. The offspring generating function is assumed to be
(f(s)=s+(1-s)^{1+\alpha}L(1-s)) for (0\le s\le1), where (\alpha\in(0,1]) and (L) is a slowly varying function at zero. Under this assumption the survival probability satisfies
(Q(n)=\mathbb P{Z(n)>0}=1-f_n(0)\sim L^*(n)n^{-1/\alpha}) and the typical size of a surviving population at generation (n) is of order ((1-f_n(0))^{-1}) (the classical Yaglom‑Slack result).

The authors focus on the rare event that the population at a distant generation (n) is unusually small. They introduce the conditioning event
(H(n,\phi(n))={0<(1-f_{\phi(n)}(0))Z(n)\le1})
where (\phi(n)\to\infty) and (\phi(n)/n\to0). This event forces the population size to be of the same order as the typical size of a process observed only after (\phi(n)) generations, i.e. much smaller than the usual ((1-f_n(0))^{-1}).

The main result (Theorem 1) describes the conditional Laplace transform of the population size at an earlier generation (m) under the event (H(n,\phi(n))). Five regimes for the relative position of (m) are considered:

1. Early generations ((m=o(n))). The conditional expectation converges to
   (\displaystyle \lim_{n\to\infty}\mathbb E!\left