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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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A random step in the $2D$ plane

Suppose we start in the Cartesian plane with coordinates $(x, y)$ such that $x^2 + y^2 < 1$ (lies in the unit circle). There are $2$ variations to my question: A step is defined as picking a random direction (not necessarily cardinal) and moving…
Arkyter
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Two points random walk

How can I build an algorithm which creates a random walk but subject to the constraint that I give the initial and final positions. How can I do it? Let's say I meet a drunk guy on the street. I know from where the guy came and also where we…
Rodolfo
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What random process is this?

This is a rather basic question, is just that I don't know a thing about this subject. Let's say $x$ is an integer. At $t=0$, the value of $x$ is 1. Then, at each time step, one of the following happens: $x$ goes to $x+1$, with probability $p_1 <…
dsign
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Expectation and variance of a biased random walk

I'm sure this should be straight forward but somehow i can't find a similar example online. How do i calculate expectation and variance in this case: A random walker takes 1 step backwards (-1) with p= 0.2, 1 step forward (+1) with p= 0.5, and stays…
LushyIvy
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Probability of reaching a point in a simple random walk on $\mathbb Z ^2$

In a simple random walk on $\mathbb Z ^2$, starting in the origin, I'm trying to find out the probability of visiting the point (1,0) before the walk returns to the origin. Means, if $O$ is the origin point, and $p$ is the point (1,0), The question…
S. R
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Average end point of 1-dimensional random walk?

Is it possible to estimate the average end point of a 1-dimensional random walk of n steps where the probability of going "left" is p and going "right" is 1-p? Thanks.
dabd
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Explanation of c# source code for random walk

In this link, the author implemented a simulation of a 2D random walk. private void RunScript(int seed, int time, ref object A) { List pList = new List(); Walker w = new Walker(); Random random = new Random(seed); …
user366312
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Does a simple symmetric random walk in an orthant always exits the orthant with probability 1?

Let $$S = \{(x_1,x_2,\ldots,x_d) \in \mathbb{Z}^d : x_i > 0, i=1,\ldots,d \}$$ and consider a simple and symmetric random walk starting in a point $x_0 \in S$. I wish to know more properties about the probability $P(x_0)$ that this random walk…
shamisen
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Does a simple random walk on $\Bbb Z^2$ admit a stationary distribution?

Can somebody answer me how I can see if a simple symmetric random walk on $\Bbb Z^2$ admits a stationary distribution? I know how to prove that this random walk is recurrent but I don't know how to see if admits a stationary distribution.
Newcomer
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Expected stopping time for a biased random walk

Let $p \in [0,1]$ and $\{X_i\}_{i =1}^\infty$ be a sequence of $\{-1, 1\}$-valued i.i.d random variables taking value 1 with probability $p$. We can consider the biased random walk $S_t$ defined at time $t$ by $$ S_t = \sum_{i=1}^t X_t. $$ Let $d >…
Harry Crimmins
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Expected conditional return time

Consider a simple random walk.Say $a$ is a vertex with only one neighbour $b$, but $b$ has various neighbours. Can someone explain why $E(Ta|X0=b)$ not equal to $E(Tb|X0=a)$ ? Where $X0$ is the position at time $t=0$ and $Ta$ is the time of the…
user756119
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Random Walks and running maximum processes

I don't know if this has been covered by other posts since those posts mention things like "Brownian motion" and "martingales", while I'm literally just in the beginning of learning this stuff. Anyway, I have 2 questions about the material in these…
Mozza
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Random Walk: $P(X = (n + k)/2) = {n \choose \frac{n + k}{2}} \frac{1}{2^n}$

My notes say the following: Example (Random Walk) A particle moves $n$ steps on a number line. The particle starts at $0$, and at each step it moves $1$ unit to the right or to the left, with equal probabilities. Assume all steps are independent.…
The Pointer
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Decision making quest. related with Markov chain or random walk(I guess?)

I'm spending much of my time solving the following question. I do understand some general strategy, which is comparing expected return for each turn, then deciding whether to go or not. But this problem is too complicated for me. Can anybody…
Joshua
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Number of paths in 1D random walk

Sorry if this has been asked elsewhere. I'd like to count number of paths that doesn't revisit zero multiple times; that is, given a starting point $a > 0$ at time $0$, what is a number of paths that will hit $b = 0$ only once at a specific time…
Tin Leelavimolsilp
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