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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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Continuous random walk

I am reading a book that is talking about continuous random walk. It first starts with defining one dimensional discrete random walk as starting at point 0 and move to either to the right or left at the rate of $1$ per unit of time. Then, it said…
mohlee
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What's the difference between a white noise process, IID process and random walk?

Just need some clarification between these concepts. Is an IID with mean zero white noise process? Also is random walk a summation of white noise process?
butterbetter
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Reading material on Random Walk on S_n using Transpositions

I am from an engineering background and I wanted to get hold of some very basic reading material on Random Walk on $S_n$ (symmetric group on n letters) using Transpositions. Could someone suggest some?
user1166419
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Probability distribution for a random walk in arbitrary dimension

I'm trying to find the probability distribution for a random walk on a lattice with lattice constant a in arbitrary dimension d. The rules for my walk is that in each step the walker has to move to an adjacent spot on the lattice along one and only…
Dargscisyhp
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Random walk in a sphere

Given a sphere of radius $R$, divided in cubic cells of size $l$, the probability for a particle to jump from a cube to another adiacent is: $P=\frac{1}{6}$. If we define the probability to exit from the sphere as: $P_0(l,R,t)$ what is the…
Riccardo.Alestra
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$\sqrt{n}$ in scaled random walk

In a reference, it is stated that $W^{(n)}(t)=\frac {1}{\sqrt{n}}M_{nt}$ with : $W^{(n)}(t)$ as scaled random walk and $M_{nt}=\sum_{j=1}^{nt}X_j$. Where does $\sqrt{n}$ come from? Would you please explain with its relationship with…
noname
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Simple Random Walk on Integers

Question concerning a simple random walk on 1D. Why the probability of hitting $\pm 2^n$ before return to $0$ is $2^{-n}$? I have no idea how to start... Thank's!
Robert W.
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Random walk on non-negative integers

Consider the Random walk on the non-negative integers with transition probabilities $$ p_{0,1}=1,~~~p_{i,i+1}=1-r,~~~p_{i,i-1}=r,~~~i\geq 1. $$ Determine $p_{00}^{(n)}$ As far as I see it…
mathfemi
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Simple random walk in the limit

Consider simple random walk: $X_n=\pm1$ with equal probabilities. $S_n =\sum_{i=1}^nX_i$. For finite $n$ we can write $$S_n=\sum_{i=1}^nX_i=\sum_{i=1}^nX_i^+ -\sum_{i=1}^nX_i^-$$ So that $$E[S_n]=\sum_{i=1}^nE[X_i^+] -\sum_{i=1}^nE[X_i^-]$$ However,…
user140541
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$1$-dimensional random walk with barrier

Let $X$ be a random walk on $\mathbb{Z}_{\ge 0}$ starting at $0$, with step size $1$, and there is a barrier at $0$ so that if one tries to move to $-1$ it stays at $0$ (non-reflecting). If we fix the number of steps, I think there is a way to…
user109870
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Random walk, discrete time, 1D, unequal discrete steps

Can someone point me towards a resource that will help me analyse a 1-d random walk where each step can take 1 of say 6 values with known probabilities. Not a continuous time random walk, time intervals are discrete and equal.
Patrick
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Stationarity conditions for $t_n = \max(t_{n-1} - \tau, 0) + a_n$

I have the following stochastic sequence $t_n$ with the following recurrent equation: $$t_n = \max(t_{n-1} - \tau, 0) + a_n,$$ where $\tau$ is known constant, $a_n$ are independent and identically distributed random variables with mean $\mu_0$ and…
0x2207
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Wilson algorithm works on $\mathbb{Z}^d$?

Suppose that we want to apply Wilson's algorithm on $\mathbb{Z}^d$, by taking a root $v_0$ and labelling the vertices of $\mathbb{Z}^d = \{ v_1 ,\ldots, v_n ,\ldots \}$ arbitrarily (using a bijection between $\mathbb{N} $ and $\mathbb{Z}^d$), thus…
3m0o
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Random walk with one absorption state, what is probability of being absorbed

I have a random walk at position i that goes to i+2 with probability 1/2 and goes to i-1 with probability 1/2. We start at position 0 and are absorbed by position -1. What is the probability of being absorbed in an infinite random walk? I am trying…
Jason
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Expected number of steps until target is reached given that we never go back to 0

I am modeling a biological process using the Moran model which effectively describes a 1D random walk. Starting at $i = 1$, the transition probabilities are: $p_{i, i+1} = \frac{ri}{ri+N} \frac{N-i}{N}$, $p_{i, i-1} = \frac{N-i}{ri+N-i}…
Dahlai
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