# Tagged Questions

For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

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### Random walk and expected value

Consider a classical symmetric random walk that starts at origin $x=0$ and lasts for $N$ steps. If $x=-1$ is reached, a walk is terminated. What is the expected value of this process? I divided a ...
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### Create a transition table

I am trying to create a transition table for a markov chain but I have difficulties. Consider a game, where each player (of two, lets call them A and B) has a fixed given probability of scoring 3 ...
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### How to use Borel-Cantelli specifically to show that the probability of a simple random walk returning to the origin in finite time is 1?

Suppose we have that $X_i$ are iid random variables with $P(X_i =1) = P(X_i = -1) = 1/2$ and that $X_0 = 0$. Then, we define the simple symmetric random walk to be $S_n = \sum_{i=1}^n X_i$. We define ...
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### For a simple random walk $S_n$ and for a stopping time $\tau$, what is the intuitive interpretation of $P(\tau < \infty) = 1$?

Suppose we have a simple random walk $S_n$ and we define a stopping time to be $\tau = min\{n: S_n = A \ \text{or} \ S_n = -B\}$. That is, we stop the first time we hit $A$ or $-B$. With this, I have ...
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### Showing a d-dimensional symmetric random walk returns infinitely often to a position that it already occupied

I have the following problem on random walks: Consider a $d$-dimensional symmetric random walk which starts at the origin at time $n=0$. Show that the walk has probability $1$ of returning ...
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### Expectation of a stopping time on an asymmetric random walk

Let $X_1, X_2, \cdots$ be i.i.d. such that $P(X_i=1)=p , P(X_i=-1)=1-p$. Denote $\tau_a = inf \; \{ n \ge 1 : S_n = a \}$ for any integer $a$, where $\tau_a = \infty$ if $S_n \neq a$ for all $n \ge 1$....
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### An example of a reducible random walk on groups?

Random walk on group is defined in the following way as a Markov chain. A theorem says the uniform distribution is stationary for all random walk on groups. If the random walk is ...
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### The expected range covered by a random walk

The question that I have been struggling with lately is: If we have a one-dimensional random walk of length $n$ (consisting of $n$ steps) with discrete steps $1$ and $-1$, with probabilities of ...
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### Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.

Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of IID random-variables s.t $\mathbb{E}\left[X_{i}\right]=0$ and $\left|X_{i}\right|\leq K$ . Let $S_{n}=\sum_{i=1}^{n}X_{i}$ , I want to show ...
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### Number of ways a dice can roll every side equally many times for the first time after x rolls

This problem is best viewed as a walk on a $d$-dimensional integer lattice with integer steps corresponding to various results of a dice roll. For a normal 6-sided dice, these would be (1,0,0),(-1,0,0)...
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### 3d symmetric random walk passes infinitely through any particular line

I'm trying to solve problem 27 from Chapter XIV An Introduction to Probability Theory Volume I by William Feller, http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/An-Introduction-to-probability-...
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### Random Walk Definition

I have just begun studying this script about Random Walks, but I'm having trouble with a definition that is given there right at the beginning (page 10). We're looking at Random Walks on the square ...
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### Random Walk Stopping Time 2

Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then $S_t= \frac{1}{t}\sum_{i=1}^{t}X_i$ is a zero mean random walk. Let $\tau$ be the stopping time corresponding to ...
Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then $S_t= \frac{1}{t}\sum_{i=1}^{t}X_i$ is a zero mean random walk. Let $\tau$ be the stopping time corresponding to ...