# Tagged Questions

1answer
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### Proof of extinction probability in Galton-Watson-process using a Martingale

this problem is somewhat similar to the thread The extinction probability of Galton-Watson process from a Martingale perspective. I want to show, that for a Galton-Watson-process $Z_0,Z_1,\ldots$ with ...
1answer
97 views

### Compute a probability in Random Walk by Martingales

Let $X_n$ be the state at time $n$ of a Markov chain with these transition probabilities : $$p_{i,i+1}=p_i\qquad,\qquad p_{i,i-1}=q_i=1-p_i$$ $(a)$ Show that $Z_n=g(X_n)\,;\,n\geq0$, is a ...
1answer
25 views

### Characterization of conditional independence

Definition: Let $\mathcal{G},\mathcal{K},\mathcal{H}$ be $\sigma$-subalgebras of $\mathcal{F}$, where $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ is a given probability space. We say that ...
1answer
107 views

### Best martingale for sequence of “dozen” bets at roulette game

Jim goes the Casino to play roulette. He only makes “dozen” bets at each spin ; his probability of winning is therefore $\frac{1}{3}$ every time (to simplify, we neglect the effect of the zeros in ...
1answer
36 views

1answer
335 views

### Non-symmetric simple random walk stopping time

Say there is a random walk $\{S_n\}$ with $S_0=0$ and $0<p=P(S_1=1)<\frac{1}{2}$. We know such a random walk would go to $-\infty$ eventually. Define the stopping time $T=\inf\{n: S_n=-\infty\}$, ...
3answers
631 views

### How to show Martingale property for sum of $S_k-E(S_k)$-summands where $S_k$ is a function of two RV's

EDIT: new formulation of the question (old version below). In a paper I found the statement that a certain sum $M_n =Y_1+\dotsb Y_n$ is a martingale, $Y_i=f (X_k, Z_k) - E ( f(X_k, Z_k) | X_k)$. (The ...