# Questions tagged [random-walk]

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

1,427 questions
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### Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
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### Is there an intuitive way to see this property of random walks?

For an $n$-step symmetric simple random walk (start at origin 0 and each step 1 unit towards left or right with equal probability,) an interesting fact is that the probability that you stop exactly at ...
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### How can I prove this bijection between random walks?

Let $R_n$ be the set of simple random walk paths such that $S_n=0.$ $P_n$ be the set of simple random walk paths such that $\forall i \in \{1,2,...,n\},$ $S_i > 0$. $N_n$ be the set of paths ...
3answers
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### Random walk on natural number

Problem: You are standing at the position $0$ on the line of natural numbers $0, 1, 2, ..., n$. From this position you go to $1$ with probability $1$, but from any other position $i$ you go to $i+1$ ...
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### Select a new value from last $N$ values; how long until the last $N$ are all the same?

Say first we have N distinct numbers in a line, like 1,2,3,...,N, in each round, we choose a ...
1answer
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### Does this modified random walk (2D) return with probability 1?

Pólya showed that a random walk (with the directions at each step uniformly distributed) on the integer lattice returns with probability 1. What if instead we consider the random walk where we are ...
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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 ...
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### Random walks and diffusion limits

Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the ...
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### Does the “prime ant” ever backtrack?

A few mathematical questions have come up from the question "The prime ant 🐜" on the Programming Puzzles & Code Golf Stack Exchange. Here is how the prime ant is defined: Initially, we have ...
3answers
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### Probability of landing on the nth stair.

Initial Question: We begin climbing a staircase beginning at stair zero. We choose to take either 1,2, or 3 steps at a time, where each number of steps have an equal chance of being chosen. What is ...
2answers
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### Biased Random Walk and PDF of Time of First Return

I have a random walk process where each step the probability of $+1$ is $p$ and $-1$ is $q$, with $p+q=1$. $p$ may not equal $q$. The walker starts at zero. I want to know the probability that the ...
3answers
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### Problem of limit with binomial coefficients

I thought that the following would made a nice exercise, but I am not sure how to evaluate its difficulty since I often miss elementary solutions. How about you try answering it? It would be great to ...
1answer
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### How to combine the four Theorems in order to prove the statement?

I have a question concerning a statement about Random Walks on $\mathbb{Z}$. Let $F$ be a distribution on $\mathbb{Z}$ which has mean $0$ and finite variance. Let $\left\{X_1,X_2,\ldots\right\}$ be an ...
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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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### Random walk on a cube

Start a random walk on a vertex of a cube, with equal probability going along the three edges that you can see (to another vertex). what is the expected number of steps to reach the opposite vertex ...
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### First player to win k matches

A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking for a probabilistic description of the outcome when looking at ...
1answer
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### Bounding survival probability of an asymmetric random walk by a symmetric one

Consider two random walks that start from point $x=0$ and time $t=0$ and move either to right $x+1$ or left $x-1$: 1) Walker 1's first move is with equal probability to the right/left. However, ...
1answer
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### Jump Process - Random Walk

A 1-D random walker strarting at time $t=0$ and location $x=0$, moves to the right ($x+1$) or the left ($x-1$) according to independent random variables $R_1,R_2,\ldots$ and $L_1,L_2,\ldots$, such ...
1answer
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### How long until everyone has been in the lead?

Earlier, I asked a question about a series of competitions: A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking ...
0answers
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### Is the following series consisting of $\pm 1$ bounded? [closed]

(Link to cross-post to MathOverflow) Let $b=\frac{\sqrt{5}-1}2$ and $a_n:=(-1)^{[nb]}$ where $[\cdot]$ denotes the floor function. Are the partial sums $A_N=\sum\limits_{n=0}^N a_n$ bounded? The ...
0answers
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### Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...