2

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ with $S_0=1$. Define $T = \min \{n: S_n=0\}$ and $$Z_k = \sum _{n=0}^{T-1} \textbf{1} \{ S_n= k, S_{n+1}=k+1 \}, Z_0 =1 .$$

Try to prove that $\{Z_k\}_{k\geq 0}$ is a Branching process ( Galton-Watson) and identify the offspring distribution.

I am not sure how can I prove this rigorously. I feel that the offspring distribution should be the geometric distribution with parameter $1/2$ but do not know how to rigorously prove it.

I got trouble in showing that $$Z_{n+1}= \sum_{i=1}^{Z_n} \xi_i,\text{if}\; Z_n>0$$ where $\xi_i$ are iid and independent of $Z_n$. I really do not know where to start in the case $Z_n>0$. I know some martingale theory but have no clue how these theory would be useful here.

Brian Ding
  • 1,866
  • 11
  • 17
  • Let $\xi^{(n)}j$ denote the number of times the random walker passes from $n+1$ to $n+2$ strictly between her $j$th passage from $n$ to $n+1$ and her $(j+1)$-st passage from $n$ to $n+1$. Then we can write $$Z{n+1} = \sum_{j=1}^{Z_n} \xi^{(n)}_j$$ – Sameer Kailasa May 27 '15 at 07:40

0 Answers0