Questions tagged [random-walk]

Question 1

First, consider a symmetric random walk $X_n := Y_1 + \dots + Y_n$, with $P(Y_k = \pm 1) = \frac{1}{2}$ for all $k \in \mathbb{N}$. For $c > 0$ define the stopping time $T_c := \min \{n \geq 0 \,|\,...

Question 2

A few years prior, an acquaintance of mine tackled a problem inspired by something in our statistics class, which basically was the idea of "what is the expected distance from the starting point ...

Question 3

Consider a symmetric simple random walk starting at $0$ and denote by $p_{n,k}$ the probability the walk occupes $k$ at time $n$. Denote also $q_n=\left(q_{n,k}\right)_{k\geq 0}$ defined by $$ q_{n,k}=...

Question 4

I am trying to rigorously derive the diffusion equation, given by $$ \frac{\partial u}{\partial t} = D\,\frac{\partial^2 u}{\partial x^2}, \qquad D = \frac{h^2}{2\tau}. $$ from a simple one-...

Question 5

Let a random walk on a graph $G=(V,E)$ be characterized by the transition matrix $P$. The usual discrete random walk process is: \begin{equation} p^{t+1}= p^{t} P, \end{equation} where $p^{t}$ is a ...

Question 6

My question came from this paper by Lalley (p.g. 2114) where he showed that the hitting/harmonic measure from finite ranged random walk on free groups is a Gibbs state. Let $(\Lambda_+,\sigma)$ be a ...

Question 7

Example 14.9 (p. 205) in Markov Chains and Mixing times describes a state space of $n$ $+$ or $-$ cards. The chain moves by interchanging two cards at random (side question: why is this a complete ...

Question 8

I start with \$1. After one iteration of a game, one of the following $m$ outcomes occurs: With probability $p_1$, my wealth multiplies by $r_1$; With probability $p_2$, my wealth multiplies by $r_2$;...

Question 9

On a given probability space $(\Omega, \mathcal{F}, \mathbb{P})$, consider the simple symmetric random walk $$ S_0 = 0, \qquad S_n = \sum_{j=1}^n \xi_j, \quad n \in \mathbb{N}, $$ where $(\xi_1, \xi_2,...

Question 10

I am looking into random walk on top of $3D$ surfaces, say along the surface of a cylinder when surface is distributed into equidistance lattice points. I can't find much of it online. Anywhere I can ...

Question 11

Consider a simple lazy random walk on 0, 1, 2, ..., $n$. It starts at $k$ and at every step gets +1 or -1 with equal probabilities $p$, or stays where it is with probability $1-2p$. Except for when ...

Question 12

Let $(X_t)_{t \ge 0}$ be a continuous-time random walk in $\mathbb Z^d$, it has an exponential holding time of rate 1 and then chooses where to jump uniformly at random from its $2d$ neighbours. I am ...

Question 13

Let $W \subseteq \mathbb{R}^2$ be a finite set of vectors, $ P$ be a probability distribution on $W$, and $V_0\in \mathbb{R}^2$ (for simplicity it suffices to consider $V_0$ where both coordinates are ...

Question 14

In Chopin and Papaspiliopoulos' An Introduction to Sequential Monte Carlo, Section 2.4, Some Examples of State Space Models, an example is given of an integrated walk in $\mathbb{R}^4$, where the ...

Question 15

I'm currently analysing a binomial random walk which is given by $$\Delta\omega_t = \chi\left(\frac{W}{N} - \omega_t\right) + \eta_t\sqrt{\gamma}\ \omega_t$$ $$\omega_{t+1} = \omega_t + \Delta\omega_t$...

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Questions tagged [random-walk]

Convergence of Random Walk and Bounded Martingals

How to derive the distribution of a 2D random walk?

References for a probability law [closed]

Derivation of Diffusion Equation in 1-D

Reversibility of a random walk

Gibbs measure is the distribution of a $k$-step Markov chain if the potential depends on the first $k$ coordinates

Markov mixing and coupling confusion: example in Markov Chains and Mixing times by Levin and Peres

Median wealth after repeated iterations of multiplicative game?

Optimal Stopping Random Walk

Random walk on surface of a cylinder [closed]

Will hitting time of a random walk increase if it starts to occasionally get lazier?

Reference for triviality of tail sigma algebra for the continuous time random walk

Expected time to exit first quadrant in 2D random walk

Why doesn't this integrated random walk admit a density in $\mathbb{R}^4$?

Analysis of a binomial random walk and calculating its median

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