Questions tagged [random-walk]
For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.
2,564 questions
3 votes
1 answer
58 views
Convergence of Random Walk and Bounded Martingals
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 \,|\,...
- 323
2 votes
0 answers
52 views
References for a probability law [closed]
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 $$ q_{n,k}=...
- 21
2 votes
0 answers
82 views
Reversibility of a random walk
Let a random walk on a graph $G=(V,E)$ be characterized by the transition matrix $P$. The usual discrete random walk process is: \begin{equation} p^{t+1}= p^{t} P, \end{equation} where $p^{t}$ is a ...
- 223
0 votes
0 answers
35 views
Markov mixing and coupling confusion: example in Markov Chains and Mixing times by Levin and Peres
Example 14.9 (p. 205) in Markov Chains and Mixing times describes a state space of $n$ $+$ or $-$ cards. The chain moves by interchanging two cards at random (side question: why is this a complete ...
- 13
1 vote
0 answers
58 views
Optimal Stopping Random Walk
On a given probability space $(\Omega, \mathcal{F}, \mathbb{P})$, consider the simple symmetric random walk $$ S_0 = 0, \qquad S_n = \sum_{j=1}^n \xi_j, \quad n \in \mathbb{N}, $$ where $(\xi_1, \xi_2,...
0 votes
0 answers
51 views
Will hitting time of a random walk increase if it starts to occasionally get lazier?
Consider a simple lazy random walk on 0, 1, 2, ..., $n$. It starts at $k$ and at every step gets +1 or -1 with equal probabilities $p$, or stays where it is with probability $1-2p$. Except for when ...
- 324
7 votes
2 answers
163 views
Expected time to exit first quadrant in 2D random walk
Let $W \subseteq \mathbb{R}^2$ be a finite set of vectors, $ P$ be a probability distribution on $W$, and $V_0\in \mathbb{R}^2$ (for simplicity it suffices to consider $V_0$ where both coordinates are ...
- 101
0 votes
0 answers
74 views
Analysis of a binomial random walk and calculating its median
I'm currently analysing a binomial random walk which is given by $$\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$...
- 395