# Questions tagged [random-walk]

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

1,406 questions
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### Unimodular walks?

I am reading a book on random walks on graphs and I found the following text: "A more general context is that of jump distributions invariant under a transitive unimodular graph automorphism subgroup ...
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### Conditional expectation for a simple random walk

Suppose that $S_n$ is a simple random walk started at $0$, so that $S_n = X_1 + \dots + X_n$ where $X_j$'s are iid random variables taking values $1$ and $-1$ with probability $p = 3/4$ and $q=1/4$ ...
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### Probability of completing a self-avoiding chessboard tour

Someone asked a question about self-avoiding random walks, and it made me think of the following: Consider a piece that starts at a corner of an ordinary $8 \times 8$ chessboard. At each turn, it ...
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### Escape time probability distribution

I have a system where a random walker is moving on $\mathbb{Z}$. However, at each point in $\mathbb{Z}$, there is a probability $q$ that an escape route exists along which the walker can escape. I ...
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### Estimating the discrete random walk probability by error function

I am trying to work out the asymptotic large $t$ behavior of following function f(t ) = \sum_{x = 0}^{2t} { 2t \choose t + x} p^{ t+x } (1 - p)^{ t - x} = \sum_{x = 0}^{2t} { 2t \...
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### Prove a general random walk is not stationary

Here is my attempt. $X_t$ is a general random walk is when we have a sequence of independent and identically distributed random variables $Y_1,Y_2,...$ such that $X_0=0,$ $X_1=Y_1$, $X_2=Y_1+Y_2$ and ...
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### Does the specific sequence of random variables converge almost surely to a given constant?

Suppose, $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d random variables, such that $P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$. Now, suppose $\{S_n\}_{n = 1}^\infty$ is a sequence of random variables ...
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### The conductance of a random walk on an undirected graph

Consider a random walk on an undirected graph consisting of an $n$-vertex path with self-loops at the both ends. With the self loops, we have $p_{xy} =1/2$ on all edges $\left(x,y\right)$, and so the ...
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### Can we relate the recurrence/transience of a lazy random walk with the recurrence/transience of a non-lazy random walk?

Consider the following discrete-time random walk on $\mathbb Z$: where at location $n\in\mathbb Z$, the walker has probability $q_n$ of taking one step left, probability pn of taking one step right, ...
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### predicting random walks with eigenvalues

(1 point) Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.6 chance you will ...
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### Hitting time on linear Markov chain

I came across this problem in a math course of mine a while back, and I haven't been able to solve it since. Anyone have any ideas? Suppose we have a chain of $n$ vertices, such that the first ...
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### random experiment with two different functions on unit interval

Let $X=[0,1]$, and functions $f(x)=x$, $g(x)=2x$ mod $1$, and the probability of chosing $f,g$: $\mu(f)=\mu(g)=\frac{1}{2}$. Now if $x$ is the starting point, then what will be a general expression of ...
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### Drunk guys in a race, who will win? (random walk)

We initially have M drunk guys located on the x-axis at positions $\mu_1,...,\,u_M$. As they are all completely "wasted", they will just randomly walk around in this 1D space for a while. After a few ...
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### Exact Probability of Collision of Two Independent Random Walkers After N Steps

Two drunks start together at the origin at $t=0$ and every second they move with equal probability either to the right or to the left, each drunk independently from the other. What is the probability ...
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### Under what conditions the gambler's ruin problem may continue indefinitely?

I'v seen the gambler's ruin problem problem on Ross.A book on probability. My question is that under what conditions on the starting money of A and B will the game continue indefinitely? In the book ...
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### Random walk on thin ice?

My Question: Is the stochastic process which is described below a (special case of a) well-studied model? What kind of properties are known under which assumptions? Let's think of a random walker on ...
Define $S_n = \Sigma_{i=0}^n X_n$, where $X_n = \pm1$ with probability $1/2$ for each case. I am trying to show that for a walk of length $2n -2k$ starting at $0$, the probability that it does not ...