Questions tagged [random-walk]

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

A random walk is a type of stochastic process with random increments, and it is usually indexed by a continuous time variable or an equally spaced discrete time variable.

An elementary example of a random walk is the random walk on $\mathbb{N}_0$, which starts at $0$ and at each step moves $+1$ or $−1$ with equal probability. The path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be approximated by random walk models, even though they may not be truly random in reality.

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Proving that $1$- and $2D$ 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 origin with probability $1$? Edit: note that while…
Isaac
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Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk: $$ E[(S_n)^2;S_{2n}=0]=\frac{n}2\,P[S_{2n-2}=0]. $$ Standard…
Did
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±1-random walk from 5 until 20 or broke

You play a game where a fair coin is flipped. You win 1 if it shows heads and lose 1 if it shows tails. You start with 5 and decide to play until you either have 20 or go broke. What is the probability that you will go broke?
koon93
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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 moves one step, either up, down, left, or right,…
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A random walk on a finite square with prime numbers

This question is following two similar questions that you can find here and here. The idea is to walk on a square of length $n\times n$, following some rules. We will identify the opposite sides. Formally, the square with the opposite sides…
E. Joseph
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Random Walk Without Repetitions

Suppose that we simulated a random walk on $\mathbb Z$ starting at $0$. At each step, we transition from position $x$ to position $x-3,\,x-2,\,x-1,\,x+1,\,x+2,$ or $x+3$ with equal probability. If we ever move to a position we have been at before,…
Milo Brandt
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Random walk on $n$-cycle

For a graph $G$, let $W$ be the (random) vertex occupied at the first time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the following remarkable fact: For the random walk on…
benny
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Expected Value of Random Walk

Can someone very simply explain to me how to compute the expected distance from the origin for a random walk in $1D, 2D$, and $3D$? I've seen several sources online stating that the expected distance is just $\sqrt{N}$ where $N$ is the number of…
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What is the probability that a random walk on $\mathbb{Z}^2$ will hit $(1,0)$ before $(2,0)$?

Suppose we have a 2-dimensional simple random walk: we start at $(0,0)$, and at every step, we add a random unit vector in one of the four cardinal directions selected independently and uniformly. It is well-known that this procedure will with…
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A prime number random walk

This question came to my mind thanks to this question which I found really interesting (and beautiful! Like the mathematician Philippe Caldero said in his book Histoires Hédonistes de Groupes et de Géométries (roughly translated) "Let us stop for a…
E. Joseph
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Random walk: police catching the thief

This is a problem about the meeting time of several independent random walks on the lattice $\mathbb{Z}^1$: Suppose there is a thief at the origin 0 and $N$ policemen at the point 2. The thief and the policemen began their random walks independently…
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Average swaps needed for a random bubble sort algorithm

Suppose we have $n$ elements in a random permutation (each permutation has equal probability initially). While the elements are not fully sorted, we swap two adjacent elements at random (e.g. the permutation $(1, 3, 2)$ can go to $(1, 2, 3)$ or $(3,…
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Probability on entering direction of a simple random walk

Let $X(n)$ be a simple random walk on $\Bbb{Z}^2$. Also we define $S_{R} = \inf\{n > 0 : X(n) \notin [-R, R]^2 \} $ : the exit time of the square $[-R, R]^2$, $T_{v} = \inf\{n > 0 : X(n) = v\}$ : the hitting time of the lattice point $v \in…
Sangchul Lee
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Probability of two people meeting in a given square grid.

Amy will walk south and east along the grid of streets shown. At the same time and at the same pace, Binh will walk north and west. The two people are walking in the same speed. What is the probability that they will meet? I tried using Pascal's…
Sarhad Salam
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Expected range of simple random walk in $\mathbb{Z^2}$

Let $(Y_k)_{k\geq0}$ be a simple random walk process. The range of an $n$-step random walk, $R_n$, is a random variable that characterizes the number of distinct points visited at time $n$: $$R_n=|\{Y_0, Y_1 \dots Y_n \}|$$ Prove that if…
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