# Branching Processes

### Fall 2012

#### M / W / F, 12:05 - 12:55 pm

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Office Hours: Wednesday 10 am - 12pm, or by appointment

Topics
• Sections: 5.4, 6.7
• Definitions:
1. Galton Watson branching process
2. Extinction
• Theorems: 5.4.1, 5.4.2, 5.4.5, 6.7.1, 6.7.7, 6.7.8
• Sample Questions:
1. Prove that if the mean offspring size is < 1, then a branching process goes extinct almost surely.
2. Say the offspring distribution is Poisson with mean 1. What is the variance of Z_n?
3. Say two offspring distributions both have mean 2, but the first has variance 1 and the second variance 2. Does one or the other have a hihger likelihood of eventually going extinct? Or is there not enough information?
4. Say the offspring distribution has mean mu and variance 1. What is the covariance of Z_n and Z_m?
5. Prove that if mu >1 there is a positive probability of never going extinct.
6. For a branching process with poisson(lambda) offspring distribution find the best asymptotics you can for the probability that the process goes extinct at step n
7. Poisson branching process with mean lambda <1 (and for lambda =1), find the asymptotics for the process surviving until at least step n
8. Derive the equation for the survival probability of a Poisson branching process with mean 2
9. Let U be uniform [0,1], and then let Z_n be a branching process with offspring distribution Bin(2, U). What is E(Z_n)?