Some Limit Distributions of Discounted Branching Random Walks

Jyy-I Hong

Department of Applied Mathematics

National Chengchi University

hongjyyi@gmail.com

We consider a Galton-Watson discounted branching random walk `\{Z_n,\zeta_n\}_{n\geq 0}`, where `Z_n` is the population of the `n`th generation and `\zeta_n` is a collection of the positions on `\mathbb{R}` of the the `Z_n` individuals in the `n`th generation, and let `Y_n` be the position of a randomly chosen individual from the nth generation and `Z_n(x)` be the number of points in `\zeta_n` that are less than or equal to `x`, for `x\in\mathbb{R}`. In this talk, we present the limit theorems for the distributions of `Y_n` and `\frac{Z_n(x)}{Z_n}` in both supercritical and explosive cases.Keyword: branching random walks, branching processes, coalescence, supercritical, explosive

References

[1] K. B. Athreya. Discounted branching random walks. Advanced Applied Probability, 17, 1985, 53-66.[2] K. B. Athreya and J.-I. Hong. An application of the coalescence theory to branching random walks. Journal of Applied Probability, 50, 2013, 893-899.