# Tagged Questions

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**2**answers

307 views

### Brownian motion of every point in the plane

Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?
What proportion of the plane does ...

**0**

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**0**answers

77 views

### Modification of a state in a random walk to be partially absorbing after a walker's position is well-approximated by a stationary distribution

Consider a random walk $(X_0, X_1, X_2, ...)$ on the interval $[0, N]$ starting from some position $k$, where $0$ and $N$ are reflecting barriers. The forward $+1$ transition probability is $p$, the ...

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**0**answers

48 views

### Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...

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**1**answer

153 views

### Quadratic variation for discrete Martingale

Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to ...

**12**

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**0**answers

505 views

### Random Walk on $\mathbb{R}$ with Uniformly Distributed Steps and “Reflective” Boundary at Origin

A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, ...

**12**

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**3**answers

431 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 ...

**5**

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**1**answer

201 views

### Memory of Uniformly Random Dyck Paths

Let $D$ be the set of all Dyck paths on square grid of size $n\times n$. For any particular Dyck path, let $S(t)=X_1+X_2+\ldots +X_t$ store the path, where $X_i=\pm 1$. Being a Dyck path, we have ...

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**1**answer

491 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 $p+q+r=1$. ...

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**0**answers

598 views

### asymmetric random walk, hitting time probability

Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$.
I am interested in the probability ...

**2**

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**1**answer

278 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 ...

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**1**answer

206 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 ...

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**1**answer

734 views

### Generating function for Random Walk Hitting Time, taking the wrong root

In a calculation of the hitting time for a Bernoulli random walk we have to calculate the hitting time $\tau(1)=\inf\{n\ge 0:S_n=1\}$ to reach $+1$ and the generating function has the recursion ...

**8**

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**2**answers

412 views

### A discrete random walk that avoids previously visited vertices for an exponentially distributed time interval

Imagine a discrete random walk on an infinite one-dimensional lattice where, for every unit interval of time, $(t_1, t_2, ...)$, the walker takes a step with uniform probability to its left or right. ...

**10**

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**2**answers

568 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 ...

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**1**answer

353 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 ...

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**1**answer

247 views

### Limit of a rescaled random sum of i.i.d. random variables

Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...

**1**

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**2**answers

239 views

### Gibbs sampling step size

I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip.
I'd like to determine that ...

**8**

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**1**answer

242 views

### Mixing time of unitary Brownian motion

Let $B_t$ be the unitary Brownian motion, i.e. Brownian motion on the unitary group $U(N)$. What is known about the mixing time of $B_t$, that is, how fast does the measure $B_t(\delta_{\{Id\}})$ ...

**2**

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**1**answer

676 views

### first passage time, brownian motion

Hi,
If X(t) is Brownian motion in 2D, where X(0) = 0, then we can ask what is the expected time required to first hit a circle of radius R, centered at the origin. This is a First Passage Time ...

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**3**answers

247 views

### How long does it take a Brownian particle to achieve a uniform probability distribution across a space?

Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" ...

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**3**answers

391 views

### Averaging over random walk on binary lattice

I have a function $f$ defined over a bit vector of length $n$. Equivalently, this is a function defined on the set of integers $[0,\ldots,2^n-1]$. I would like to compute the mean or variance or some ...

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**1**answer

626 views

### How far can a particle travel from its origin if it exhibits self-avoiding Brownian motion in two-dimensions?

Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously ...

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**1**answer

220 views

### Self Avoiding Walk Pair Correlation

Let $\gamma(i)$ be a self avoiding walk (SAW) on a 2D lattice $L$ (a square lattice for example) starting at a predefined origin ( $\gamma(0)=(0,0)$ ) and having length $n:=\ell(\gamma)$. Furthermore, ...

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491 views

### Random walks on graphs: Cover time and blanket time

Winkler and Zuckerman conjectured that the blanket time is within a constant factor of the cover time. The conjecture was recently proved. The cover time $C$ is the expectation of the first time $t$ ...

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**2**answers

986 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
...