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 nth generation and \zeta_n is a collection of the positions on \mathbb{R} of the the Z_n individuals in the nth 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.