מושגי ליבה
The authors establish quantitative bounds on the rate of convergence of the Yaglom limit for critical Galton-Watson processes in a varying environment, using the Wasserstein distance.
תקציר
The key points are:
Galton-Watson processes in a varying environment (GWVE) are discrete-time branching processes where the offspring distributions vary across generations.
In the critical case, these processes have a Yaglom limit, where a suitable normalization of the process conditioned on non-extinction converges in distribution to a standard exponential random variable.
The authors provide explicit bounds on the rate of convergence of this Yaglom limit, measured in the Wasserstein distance.
They consider two different normalizations - one using the mean of the conditioned process, and one using the product of the mean and the normalized second factorial moment.
The bounds depend on the growth rates of the mean and normalized second factorial moment sequences, as well as some technical conditions on the offspring distributions.
The authors also provide examples illustrating the sharpness of their bounds in certain cases.
The proofs rely on Stein's method for exponential approximation, along with a careful analysis of the size-biased distributions associated with the GWVE.