All Questions
24 questions
14
votes
2
answers
2k
views
Markov chains: invariant measures and explosion
The following seems like such an elementary question, but I didn't get anywhere with it.
Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure (...
6
votes
2
answers
2k
views
Random walk to stay in an interval forever
Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities.
To ...
- 502
6
votes
1
answer
170
views
Basic Definition and Notations in RWRE
From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...
- 61
5
votes
0
answers
485
views
Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)
Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
- 1,127
5
votes
0
answers
95
views
Most visited vertex in a random walk with place dependent drift
Consider the following Markov chain on $\mathbb{Z}$:
$$
P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}}
$$
Do there exist constants $c,C>0$ such that
$$
c\cdot P^t(z,z) \...
- 527
4
votes
1
answer
176
views
Random Walk with "Forward Dependency"
Let $\{X_t\}_{t=-\infty}^{\infty}$ be a sequence of random variables. We are interested in a "random walk" (or more generally, a random field) that can be characterized by
$$
X_t ~|~ X_{t-k}, \ldots, ...
- 1,127
4
votes
1
answer
518
views
Probability that two walkers will meet on a graph
Consider two independent continuous random walks on a graph $G$ with adjacency matrix $A$. I am interested in the probability that the two walkers will ever meet.
When the graph is a $k$-regular ...
- 117
4
votes
2
answers
255
views
The necessary sufficient condition for recurrence of a Markovian random walk
Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...
- 41
3
votes
1
answer
2k
views
Hitting time probability in a Random Walk with possibility to die.
A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...
- 295
3
votes
1
answer
340
views
Importance resampling with exponential weighting
Suppose that we have
$$
\frac{p(x)}{q(x)} \propto \exp(\tau f(x)),
$$
where we can sample from $q$ but not from $p$. Our goal is to generate a set of particles $\{x_i\}_{i=1}^n$ such that $n^{-1}\sum_{...
- 1,127
2
votes
1
answer
412
views
Does random walk have more concentration surrounding the origin?
Consider a simple random walk $S_n$ on one dimension, starting at $0$. In this case, $S_n$ fluctuates between $-\infty$ and $\infty$, but intuition says that it might stay more often in an interval ...
- 502
2
votes
1
answer
168
views
Random Walk 2D with dependent weights [closed]
I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (...
- 23
2
votes
1
answer
421
views
Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
- 340
2
votes
1
answer
447
views
MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
1
vote
1
answer
4k
views
First passage time of a 1D simple random walk in a discrete time infinite markov chain [closed]
If we consider a simple Random Walk on the positive integers (discrete Markov chain), with symmetric transition probabilities. We start at time $0$ at the integer $i_0 = m$ and at each time step $P(...
- 53
1
vote
2
answers
302
views
how to derive stationary distribution of maximal entropy random walk
I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps.
Description:
The ...
- 211
1
vote
1
answer
233
views
Random walks on Galton–Watson trees
I am working on a paper of Elie Aidekon : ‘Speed of the biased random walk on a Galton–Watson tree’ and have a question about one transformation in a proof:
\begin{align}
& 1+\frac{1}{1-\lambda}+\...
- 39
1
vote
1
answer
276
views
Number of deaths in birth-death process conditioned on start and end points
Say I have a simple linear continuous time birth-death process with state space the non-negative integers, where there are parameters $b$ and $d$, with the rate (as you'd see in a $Q$ matrix) of going ...
- 11
1
vote
0
answers
41
views
Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
1
vote
0
answers
181
views
Random walk on 2d lattice with obstacles
Consider a random work on $L=\mathbb Z^2$ endowed with obstacles (i.e each cell $(x,y)$ of $L$ may contain a obstacle, i.e the random walk halts whenever it hits such a cell). Let $P(x,y) = 1$ if cell ...
- 6,853
0
votes
1
answer
110
views
Skip-free random walks: recurrence and transience
Let us define a one dimensional random walk: for all $n\in\mathbb{N}$
$$
X_n:=\sum_{i=1}^nZ_i
$$
with $Z_i$ i.i.d. random variables taking values in $\{-1,0,1,2,\dots\}$. This process is sometimes ...
- 59
0
votes
1
answer
613
views
2 Random Walkers on 2d square lattice, Torus
I am looking for the probability that two random walkers initially at different sites, meet at step t if they are moving on a 2-dimensional torus(Square Lattice)
Any help would be appreciated.
- 53
0
votes
0
answers
85
views
Does a 2d random walk hit 0 for increasing distances AND time spans?
Question: For a simple symmetric random walk $(Z_t)_{t\geq 0}$ in $\mathbb{Z}^2$, does
$$\lim_{\beta\rightarrow 0}\mathbb{P}^{x_\beta}(Z_t=0\text{ for some }t\leq h(\beta)T)=0\quad (2.8)$$
where $|x_\...
- 31
0
votes
0
answers
111
views
Markov chains on a polyhedron
A modification of a question from Gerard Letac (1976): A m-sided q-adjacent-faced polyhedron has one of its faces "up." Each round, the polyhedron rolls so that any of the adjacent faces is now up. ...
