# Questions tagged [random-walk]

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

1,424 questions
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### Is there a deeper reason why the simple symmetric random walk on $\Bbb Z^D$ turns transient when increasing $D$ from 2 to 3?

Polya proved the following very well-known Theorem: A simple random walk on $\Bbb Z^D$ is recurrent if and only if it is symmetric and $D\le2$. Dropping simplicity (i.e. allowing jumps to non-...
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### Prove that every position is equally as likely in this random walk scenario

There are two points $A$ and $B$. You are standing in the middle between them. In each step, go half the way to the point $A$, or half the way to the point $B$, each with probability of $0.5$. Mark ...
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### Simple Probability Matrix

Question: Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will ...
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### Drunkard’s walk Markov chain

Consider the Drunkard’s walk Markov chain with state space $X =${$0, 1, . . . , N$} and transition matrix: where $0 < α < 1$ is the probability of moving one step from position $k$ to position ...
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### Reference/suggestion needed - Hitting time of domain equal everywhere -> {domain = sphere, x0 = centre}

So, given a Brownian motion under drift $dX(t) = c dt + dB(t)$ where B(t) is a std Brownian process. Let the process start within some (convex, etc) domain $A$, i.e. $x_0\in A$. Does the following ...
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### Random walks in triangular and hexagonal lattices

I'm trying to show that the random walk on the triangular and hexagonal lattices in the plane is recurrent, but I'm struggling to find a theorem I could use to prove this, and how to proceed in ...
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### Upper bound on the number of fixed length, self avoiding paths between two points on a 2D grid

Given some point $x$ on the 2D grid $\mathbb{Z}^2$, is there an upper bound on the number of self avoiding walks of length $n$ between the origin and this point? I have looked at some literature on ...
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### Stationary distribution of “almost” a random walk

Consider a random walk on $\{1,2,\dots,N\}$, that in state $i$ has probability $p_i$ of going left/right and probability $1-2p_i$ of staying in place, so the transition matrix is \begin{align*} \...
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### Maximum of a zero-mean random walk

Assume $S_n=\sum_{k=1}^n X_k$ where $X_k$ is i.i.d distributed and $\mathbb{E}X_1=0$ and $\mathbb EX_1^2<\infty$. Let $M_n=\max_{1\leq k \leq n}\{S_k\}$. What is the exponential asymptotic ...
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### MATLAB - Least Squares Fitting for Log - Log Data to find p value.

This is part of the code for a random-walk simulation. To test the code, I'm using steps=[30]; there will be more values, but I want to run it for 1 trial to decrease code processing. ...
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### MonteCarlo Random Walk Simulation - steps should be scaled by tmp ? (MATLAB)

I have a question regarding this code snippet that we changed for project 3 Monte-Carlo & Random Walk. monte-carlo-code-segment The code was changed to process all the stepNs. it loops through a ...
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### Random Walk leaves the compact set about $0$

Let $(X_{k})_{k}$ be IID random variables on $(\Omega, \mathcal{F}, P)$ where $0 < \mathbb E[|X_{1}|]<\infty$ and $S_{n}:=\sum_{k=1}^{n}X_{k}$ Show that $(S_{n})_{n}$ leaves the compact set ...
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### Chance of not finding an alternating Hamiltonian path in a colored graph in $n$ random walks

Introduction Given a random graph which can be constructed as follows: generate $s$ pairs of unconnected black edges and $2 \cdot s$ vertices connect these black edges using red edges, where the ...
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### How to calculate a number of all possible random walks without self-intersections?

Is there a certain formula, that gives number of all possible non-self-intersected N walks? N is number of steps of the walk. ...
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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 ...
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### Expected number of visits to $k$ before hitting 0

This problem is from Exercise 5.5.6 in Durrett's Probability: Theory and Examples, 5/E, Use Theorems 5.5.7 and 5.5.9 to show that for simple random walk on $\mathbb{Z}$, if we start from $k$ the ...
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### Probability of random walk in a specific point - 2D Random Walk -

I've written a simulation in matlab of a 2-D random walk that, at any point, has an equal probability of going to any of the adjacent points. The simulation was run for 10,000 steps on a grid with ...
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### Conditional Probability and Conditional Independence in Random Walk

A robot randomly walks backward and forward, with probability $p$ and $1-p$ respectively. Let $S_i$ denote the direction he walks on the $i$-th step. Are $S_1$ and $S_2$ independent? What assumptions ...
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### Simulating a continuous-time, discrete-state Markov chain in fixed time step

To simulate a continuous-time, discrete-state Markov chain with known transition probabilities, we can generate exponentially-distributed waiting time according to current total transition rate, and ...
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### 1D Random walk with reflective barriers — average time to return to origin

I have very limited probability background, but I came across a problem in an engineering application: Is there a formula that computes the average number of steps taken for a particle beginning at ...
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### A probability concerning the maximum and minimum of a simple random walk

Let $X_i$ be i.i.d. such that $\mathbb P(X_i = 1 )=\mathbb P(X_i=-1) =\frac1 2$. Let $a\in \{1,2,....\}$, now define the random walk, $S_0=a$ and $$S_n = a+\sum_{i=1}^n X_i$$ Now define the maximum ...
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### Expected value of random walk with optional stopping

I want to compute the expected value of the following game. I have a symmetric 1-D random walk, I can choose to stop at any time and my winnings will be the value of the random walk. What is the value ...
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### Exponential ergodicity of biased random walk confined to the positive integer quadrant

I am looking at a discrete-time random walk on $(\mathbb{Z}^{+})^n$, where $\mathbb{Z}^{+}:=\{0,1,2,\dots\}$ and $n\in\mathbb{N}$. At any time, the random walk chooses a uniformly random co-ordinate ...
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### Expectation of fourier exponential

Suppose I have some probability distribution in $\mathbb{R}^n$ and I want to calculate the expectation value of $$\left\langle e^{-i\vec{q}\cdot\vec{r}}\right\rangle^n.$$ My professor says that the ...
Let $(S_n)$ be an elementary random walk, ie. $S_n = \sum_{i=1}^n X_i$ where $P(X_i = 1) = P(X_i = -1) = \frac12$. Let $T = \inf \{ n : S_n \in \{-2,2\}\}$. $T$ is clearly a stopping time and is ...
Is the random walk on an isoradial graph defined in Chelkak and Smirnov (2011) (in pages 3-4) transient? Let us define the random walk on an isoradial graph $\Gamma$ starting from $x$ by, \begin{...