All Questions
Tagged with stochastic-processes random-walks
12 questions
12
votes
3
answers
666
views
An "inchworm-like" random walk on an integer interval
Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the ...
- 121
11
votes
2
answers
1k
views
Is there a differentiable random walk?
Is there a random walk which is differentiable or smooth? Like brownian motion except smoothed out on small distances. I was wondering if there is a "natural" or "canonical" analogue of brownian ...
- 111
10
votes
4
answers
679
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 773
8
votes
2
answers
259
views
Particularities about the honeycomb lattice for the computation of connectivity constant
After reading the paper The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$ by Hugo Duminil-Copin and Stanislav Smirnov (arXiv:1007.0575) published some time ago in Annals Math....
- 1,561
5
votes
1
answer
523
views
Scaling of First-passage times for Random Walk on integer lattices
Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...
- 61
4
votes
1
answer
829
views
Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
- 141
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
1
vote
1
answer
385
views
How fast does this Gaussian random walk move away from the origin?
Suppose $z_i$ are IID zero-centered $d$-dimensional Gaussian random variables with unit-trace covariance $\Sigma$ and $g(z_i)$ is the sum of its components.
Consider the following random walk:
$$x_s=\...
- 2,442
1
vote
1
answer
247
views
Arc Sine law for Random Walk conditioned to non-absorption or not?
Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$.
Is the $E(S^{2}_{n}| \tau \geq n)$ known?...
- 61
1
vote
1
answer
456
views
Random walk with gaussian increments - Probability that it falls below 0
Suppose $\{Z_{i}\}_{i=1,2,\ldots}$ are normally distributed (identically and independent) random variables with mean $\mu>0$ and positive variance $\sigma^{2}$. Suppose we want to calculate the ...
- 13
0
votes
1
answer
183
views
Probability to cross dynamic boundary for 1D-random walk?
context: Imagine we have an evolving bit sequence (ex: 001011...) where the probability to get 0 or 1 is 1/2. n is the lengh of my sequence (the number of bits)
I can make an analogy with random walk: ...
- 5
0
votes
2
answers
3k
views
Two dimensional brownian motion first passage time
Hello,
I am looking for information on how to solve/compute first passage time for two dimensional Brownian motion.
any papers, references, books or web links for study will be helpful.
thanks
...
- 21