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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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Infinite number of 1D-random walkers

Place exactly one random walker at each integer in $\Bbb Z$ and define $Y_n$ as the number of these who are at the origin at time n. Show that $0<\displaystyle\lim_{n\to\infty}P\{Y_n=0\}<1$ and find $\displaystyle\lim_{n\to\infty}P\{Y_n=k\}$ for…
user1976
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Random walk with gaussian increments - Probability that it falls below 0

Suppose $\{Z_{i}\}_{i=1,2,\ldots}$ are normally distributed (identically and independent) random variables with mean $\mu>0$ and positive variance $\sigma^{2}$. Suppose we want to calculate the probability, that $$U(n)=u+\sum_1^nZ_i\le0 \text{ for…
Babypopo
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Continuous time random walk

I am simulating Continuous time random walk in order to simulate anomalous diffusion. (python code) x= np.zeros((time)) y= np.zeros((time)) xs = 0 ys = 0 ts = 0 alpha=0.5 for t in range(time): while ts < t: alpha = 0.5 #Here I…
Jean-m. a.
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Survival probability of 1D Random Walker

For a 1 D random walk on $Z$ axis, starting at $z=0$, equal probability to go to right or left, what is the probability that during the first k steps the walker's position remains $z\leq m$? This is also called the survival probability. I can write…
Albert
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Expectation of a Random Walk

I am researching Random Walks and trying to find how to get their expectations. I have studied Markov chains before. I have found one way of getting the expected number of steps to reach a state by using generating functions. Can anyone tell me if…
JJJ
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Random Walk with Edges

The setup for the specific problem that led to this question is as follows: You are playing a game at a casino and have \$10,000; The bank has \$2,000. You are making \$1,000 bets, with a equal payout. You play until either you or the bank run…
AJMansfield
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A novel random walk problem

I'd like to propose a novel type of random walk problem. Suppose we are looking at a 1D discrete time random walk. Suppose we restrict the movement of the walker in the following way: If the walker has just taken two steps to right in the previous…
Chris
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Random walk on circle equivalent to random walk on line with reflecting boundaries

I am looking at (1D) random walks in discrete time and space, where a fair coin determines the direction the walker moves in. I am interested in the probability of the walk to return to the origin at every time step. I found a question on here that…
Q.Ask
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Approximating a correlated random walk on a 2D grid

I have been working on modeling the movement of ants, and a number of studies suggest the use of correlated random walks. These are biased random walks, where the direction of the next step is dependent on the direction of the previous step. The way…
Tejas
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symmetric simple random walk: P(Si≥0 for all 1≤i≤2n | S2n=0)

The question is $\mathbb{P}$($X_{i}$ =1)=$\mathbb{P}$($X_{i}$ = -1)=1/2 for all 1≤i≤2n. For every 1≤i≤2n, Define $S_{i}$=$X_{1}$+$X_{2}$+...$X_{i}$. What is the conditional probability $\mathbb{P}$($S_{i}$≥0 for all 1≤i≤2n | $S_{2n}$=0)? And my…
Charlie F
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Generating Functions of 1D Random Walk

Define $\displaystyle F(s) = \sum_{n=0}^\infty f_ns^n$, where $f_n= P($ the first return to the origin at time $n)$. Suppose $F^{(n)}(s) = \displaystyle \sum_{k=0}^\infty f_k^{(n)} s^k$, where $f_k^{(n)}$ denotes the probability of the $n$th…
CharlieCornell
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Random walk: Need intuition behind the expected distance from origin versus expected deviation of the fraction of left/right steps.

It appears that in a 1D random walk the expected RMS distance from the origin is some positive value with any positive number of steps (N): sqrt(N)/2 However, in this same walk, the expected deviation of the fraction of steps to any one direction…
imagineerThat
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Centered random walk on $\mathbf{Z}^2$ is recurrent?

Is the random walk on $\mathbf{Z}^2$ started at $\left\lbrace \left(0,0\right) \right\rbrace $, in which the probability of advance to right is $p$, the probability of retreat to left is $p$, the probability of advance to up is $q$ and the…
Carlos Andres Henao Acevedo
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Random walk probability of first arriving at some value

If I start a random walk at (a, b) in the x-y plane (say, 5, 5), how do I calculate the probability that the first time I hit the y axis, I land on a positive y value? I am not sure how to go about calculating this -- I just read the Wikipedia…
rxmnnxfpvg
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Hitting time definition in “Quantum Walks via Linear Algebra”

In the book “Quantum Algorithms via Linear Algebra” I found this definition for the hitting time of a random walk: $k$ giving $max_{u,v}|D(v) - A^l[u,v]| < \epsilon$ for all $l \geq k$ is called the hitting time. Where $A$ is the walk matrix and…
x squared
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