# 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 process theory: probability distribution of height vs summits

Imagine I have a matrix of height values ($z$), e.g. a surface height topography. This surface is a random process: randomly rough isotropic surface with Gaussian distribution. What is the difference ...
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### Random solving of a Rubik cube .

After playing a little with a Rubik cube I thought of the following problem : Suppose we start with a solved Rubik cube (a general one , with $n^3$ cubes) . Then we choose one of the moves , each ...
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### Probability a random walk eventually crosses a square root boundary

Let $\lbrace X_n, n \geq 1 \rbrace$ be i.i.d random variables taking values in $\lbrace -1, 1 \rbrace$, and \begin{align*} S_n = \sum_{i = 1}^{n} X_i \end{align*} be a random walk. Let $f$ be a ...
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### Simple random walk on the $N$-cycle

I am considering the following example: In my lecture notes we noted that "the functions $(\phi_j)_j$ form a basis". I think they refer to the space $\mathbb{C}^G$ where $G$ is the above ...
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### Upper bound for random walk to show stopping time is bounded

I have a simple symmetric random walk (SSRW), and a stopping time: $\tau=\inf\{ n \geq 0 ~:~ |S_n|=N\}$. I am showing that $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...
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### Bound on Poisson process

In a proof of a theorem I have the following situation: $N_t$ is a Poisson random variable with parameter $t$. From a corollary we get the following result: Let $X_1,X_2,\dots$ be independent, ...
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### What is the probability a random walk crosses a line before another?

Let $n \geq 0$, $X_n$ be a random walk, where $X_{n+1} = X_n + 1$ with probability $p$, and $X_{n+1} = X_n - 1$ with probability $1-p$. $X_0 = 0$ Let $l_n, r_n$ be a sequence of integers, where for ...
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### Applications of Random Walks for undergraduate students

Students are asking for applications of discrete random walks in "real life" problems. By real life they mean financial applications and industry. We have two more weeks on this subjects and I'm ...
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### Triangular inequality for n-th step distributions

Assume that $p_n$ is the $n$-th step distribution of a random walk with state space $\mathbb{Z}^d$, i.e. $p_n(x,y)=\mathbb{P}(S_{n+1}=y\mid S_0=x)$, where $S_n=S_0+\sum_{i=1}^nX_i$ with $X_i$'s i.i.d. ...
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### Gambler's Ruin with no set target for win

I have been presented with the following probability question: A compulsive gambler is never satisﬁed. At each stage he wins $€1$ with probability $p$ and loses $€1$ otherwise. Find the probability ...
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### Number of random walk paths crossing horizontal line

I have series of binomial variables $\xi_1, \xi_2, \dots, \xi_n$ which form a random walk. Variables can be $\pm 1$ with probability $\frac{1}{2}$ and we define $S_n = \sum_{i=1}^n \xi_i$. We start ...
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### Eigenvector/value of a biased random walk with a sink and a wall

Suppose you have a one dimensional random walk, with a wall at $S=0$ and a sink at $S=n$. The walk is biased so the odds of moving down vs moving up are $b:1$. More concretely, the transition graph is:...
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### Expected steps for ant on a cube, [duplicate]

There is an ant on a vertex of the cube, he's trying to get to the opposite vertex, what's the expected steps for it to take before reaching the opposite vertex? Ant can move in any directions along ...
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### Expected number of steps for a random walk- robot

A robot is located at the top-left corner of a m x n grid The robot is trying to reach the bottom-right corner of the grid, he can move randomly in any of the directions: up, down, left, right. ...
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### Properties of a random walk [closed]

First of all, I know nothing about Markov chains, and I'd like to prove the following without using the theory around them. Let $(M_{n})_{n\geq 1}$ be a random walk over $\mathbb{Z}$, starting at \$...