# Tagged Questions

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### Proof that random walk visits zero infinitely many times

Since the Green function $G(x,1)=\sum\limits_{n\in \mathbb{N}_0}P(S_n=x), x\in\mathbb{Z}^d$ gives the expected number of visits to $x$ in a random walk, I'm asked to prove the following: I have to ...
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### One dimension symmetric random walk

We have person that starts at $x=0$ and at every step he goes left with probability $0.5$ and right with probability $0.5$. What is the probability he will arrive at $x=3$ at some time? I got ...
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### Random walk on a finite square grid: probability of given position after 15 or 3600 moves

An ant is walking on the squares of a 5x5 grid - it starts in the center square. Each second, it will choose (with equal probability) to do one of the following: Move north one square Move south ...
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### Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $P_{k,j}$ be the probability that the walker hits the point $k$ without returning to the origin in ...
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### Intuition in Random walk

Suppose $X_i$ are i.i.d. r.v. $S_n=X_1+\cdots+X_n$ is random walk. Why $\mathcal{F}_n =\sigma(X_1,\cdots,X_n)$ are called the information known at time n? I think We only know the measurability of ...
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### random walk with dependent increment

Consider the following sort of random walk. The position of the walker at time $t$ is represented by the random variable $r(t)$, with $r(0) = 0$. The variable satisfies the following equation,  ...
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### Random walk with weighted probabilities

Taking a walk on $\mathbb{N}$, starting at 1, I need to find out how many steps I expect to take before returning to the origin, as a fraction. For each step, I either walk forward, backward, or stay ...
So I have a series: $f(x_{n+1})=x_n \pm t$ and $f(x_0)=W$ What I'd like to calculate is the probability in terms of $t$ and $W$ (assuming they're any constant $W>t$) that any $f(x_q)=0$ for all ...
Let a particle in the plane $R^2$ executes random jumps at discrete times $t= 1, 2, ...$. At each step, the particle jumps from the point it is a distance of lenght one. The angle of any new jump ...