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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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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…
Diego
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height of domino tower

Suppose you are building a domino tower using identical pieces of unit length. You place a new domino piece, one at a time, on the top of the tower. However there is a random error in the placement of dominoes, so a new piece is placed some small…
Maxim Umansky
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Random walk on $\mathbb{R}^2$

Question Consider a random walk on the plane $\mathbb{R}^2$, a point randomly chooses an angle $\theta$ at each step and walks forward one unit length. Then after $t$ steps, what is the probability that this point is within a unit length from the…
Aster
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If I wanted to randomly find someone in an amusement park, would I be faster roaming around or standing still?

Assumptions: The other person is constantly and randomly roaming Foot traffic concentration is the same at all points of the park Field of vision is always the same and unobstructed Same walking speed for both parties The other person is NOT…
Dargscisyhp
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Self-avoiding random walk on $\mathbb{Z}^2$ getting stuck

Let $W_n$ be a self-avoiding random walk (SAW) on $\mathbb{Z}^2$, starting at the origin. Formally, $W_0=0$ and for $n\ge 0$, $W_{n+1}$ is chosen uniformly from the neighbours of $W_n$ which were not visited earlier: $$W_{n+1}\sim…
Bach
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Modelling the meeting of two random walks

Consider the following situation: You have two lovers that are drunk and are randomly walking around on a graph. Once they meet, they will continue randomly walking around as a single pair. How can I model this situation mathematically? I can…
user17793
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Random walk on $\mathbb{Z}$ with more than two possible steps

Let be $\{X_n\}_{n\in \mathbb{N}}$ random walk on $\mathbb{Z}$. Let be $$P(X_{n+1} = k + a| X_n = k)= p_a$$ for $a\in \mathcal{A} \subset \mathbb{Z}$. Let say that $X_0 = 0$. I am interested in probabilities $$\tilde{p}_k = P(\exists n\in…
Antoine
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2-dimensional random walk

I have a question which I anticipated to be rather easy initially. After some googling, however, I realized it is actually not that easy. It concerns a 2-dimensional random walk with constant unit step-size,…
Kees
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random walk on real line

Suppose I start at $A>0$ and every period I either move a distance $B$ to the right with probability $p$ or a distance $C$ to the left with probability $1-p$. The expected move is positive: $p\times B+(1-p)\times C>0$. Every period I die with…
pyanni
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Symmetric random walk with bounds

can anyone help me with this: We are considering a symmetric random walk that ends if level 3 is reached or level -1 is reached. Start=0 What is the expected number of walks? So I am looking for: $E[{\tau}]$ with $\tau$=the stopping time.
user59871
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Biased random walk in 1D - expected hitting time for either edge of box

I have a random walk process, discrete in time and state, where at each step the probability of $+1$ is $p$ and $−1$ is $q$. $p+q=1$ and $p$ may be different from $q$ (i.e. the random walk is "biased", "asymmetric", has "drift"). Starting from $x$,…
Dan Stowell
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Drunkard walk with payments: when should the drunkard pay?

A drunkard stands at the origin. Each second he goes right with probability $p$, left with probability $q$, and remains still with probability $1-p-q$. When he arrives at $x=+D$ or $x=-D$, he pays $M$ dollars and is transported back to the…
Erel Segal-Halevi
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Why does infinite expected number of returns to the origin imply a random walk returns to the origin with probability 1?

In proving that a simple symmetric 2-d random walk a.s. returns to the origin, the proofs generally start by showing (*) that the expected number of returns to the origin is infinite, and then use a lemma that: Lemma 1: If that expected number of…
Mark Fischler
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Random walks and their uses

Can anyone provide some motivation behind the use of random walks? I know they're used a lot in computer science, in things like page walk (I think that's what it was called- something like pagerank), but they are also used to model things like…
user82004
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Existence of a stationary distribution for a random walk

Consider a random walk on a infinitely countable connected graph. We assume that each vertex has finitely many neighbors and that we have a uniform bound of the number of neighbors at each vertex. The probability to move from x to a neighbor y of x…
Link
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