Lagmental Vicfred

A Branching Process Goes Extinct at a Fixed Point by Vicfred

tags: mathematics, martingales, branching processes, optional stopping

The extinction probability is the smallest fixed point of the offspring generating function. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

A process \((M_n,\mathcal F_n)\) is a martingale when \(\mathbf E[M_{n+1}\mid\mathcal F_n]=M_n\). Its baseline \(M_0\) models fair evolution relative to the information currently available.

$$ f(s)=\mathbf E[s^\xi]=\sum_{k\ge0}p_ks^k $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ q=\mathbf P(\text{eventual extinction})=\min\{s\in[0,1]:f(s)=s\} $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ q=\begin{cases}1,&f'(1)\le1,\\<1,&f'(1)>1,\end{cases}\qquad f^{\circ n}(0)=\mathbf P(Z_n=0)\uparrow q $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(s)=\mathbf E[s^\xi]=\sum_{k\ge0}p_ks^k,\\[5pt] \mathsf{C}\;&:\quad q=\mathbf P(\text{eventual extinction})=\min\{s\in[0,1]:f(s)=s\}. \end{aligned} $$

A nearby false statement

Optional stopping is not valid for every stopping time. Boundedness, integrability, or uniform-integrability conditions prevent hidden mass from escaping at infinity.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] q=\mathbf P(\text{eventual extinction})=\min\{s\in[0,1]:f(s)=s\} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Mon 26 July 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.