1
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1answer
36 views

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 ...
5
votes
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 ...
1
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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 ...
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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 ...
2
votes
1answer
36 views

Is the following Markov Chain a martingale?

Say I have a finite, ergodic Markov chain with states ${0,1,2,3}$ and with the following transition matrix: $$\begin{bmatrix} \frac{7}{10} & \frac{3}{10} & 0 &0\\ \frac{1}{10} & ...
0
votes
0answers
113 views

understanding submartingale proof with discrete state space

I am reading a text about branching markov chains: My question is about the first half of page 8 where $Q(t)$ is proven to be a submartingale. Briefly the used notation: $t$ is discrete time, $n(t)$ ...
0
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1answer
2k views

Stochastic process that is Martingale but not Markov? [duplicate]

Can you please help me by giving an example of a stochastic process that is Martingale but not Markov process for discrete case?
1
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1answer
229 views

Asymmetric random walk with unequal step size other than 1.

Say, an asymmetric random walk, at each step it goes left by 1 step with chance $p$, and goes right by $a$ steps with chance $1-p$. (where $a$ is positive constant). The chain stops whenever it ...
0
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0answers
77 views

Martingale with reflecting barrier

I am not very familiar with the theory of martingales or random walks, perhaps someone could point me in the right direction or give me some help with the following problem. Consider a random ...
4
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1answer
459 views

Markov chains- recurrence and transience

This is an exercise in Durrett's probability book. $p$ is the transition probability for a markov chain on a countable space. $f$ is said to be superharmonic if $f(x)\geq\sum_y p(x,y)f(y)$, or ...
3
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2answers
214 views

Martingale associated to Markov chain

$X$ is a (continuous time) Markov chain with generator matrix $\Lambda$ and finite state space $G$. I know that for $g\colon G \to R$ $$ M_t = g(X_t) - g(X_0) - \int_0^t (\Lambda g)(X_s)\, ds $$ is a ...
2
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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 $T=\inf\{n: S_n=-\infty\}$, ...
2
votes
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 ...