Admistrivia

• Midterm next week
• Questions from the homework?

Chapter III.8: Branching processes

Examples

• electron mutipliers
• Neutron chain reaction
• Family names
• Biology: genes
• Biology: bird flu

Model

• Let B_i be the number of births a person has
• Each generatino is exactly one life long
• X_n+1 = sum_i=1,Xn B_i

Mean growth

• let mu = E(B)
• E(X_n+1|X_n) = B X_n
• So M(n) = E(X_n) = muⁿ if X₁ = 1

Variance

What if mu = 1???

Extinction probability

Theorem: probability of extinction = 1. Why? Martingale maximal inequality.

What if mu < 1? Use coupling. Give each family some extra virtual children. Now mu = 1. The whole population goes extinct, hence, so does the real population. (cute coupling examle)

What if mu > 1?

First step analysis

• Let u_n = probability of extinction in n steps starting from X = 1.
• u_i^k = starting from X = k.
• First step equation is then easy: u_n = sum p(k) u_n-1^k

Limiting behavior

• u is increasing, and less than one.
• Hence limit exists.
• Suppose limit is one. Then we can do taylor series.
• ... which shows mean is less than or equal to one.
• Theorem: mean <= 1 iff extincion is inevitable.
• See section III.9 for martingale free version of this proof.