# Questions tagged [random-walks]

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### Discrete random walk on polytope via involutions

Let $P$ be a convex polytope (or more generally convex body, I suppose) in $V=\mathbb{R}^n$. For each $v\in \mathbb{P}V$, we define an involution $\tau_v\colon P\to P$ by setting $\tau_v(p)$ to be the ...
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### Random walk always stays below a level $a$

Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that $$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$ where $\mu$ is close (or going) to zero. We also assume that the moment ...
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### A random process with conserved momentum: 'particle decay'?

Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...
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### 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}+\...
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### Symmetric random walks - bounds on the amount of time spent in a subset $A$?

For a symmetric random walk $S_n$, if we have a bound $P(S_n \in A) < \delta$, do we get a bound on the amount of time $S_n$ spends in $A$? Let $S_n$ be a symmetric random walk on the integers. ...
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### Occupation time of non-stationary random walk

Assume $\varepsilon \in [0,1/2]$. Consider the discrete-time random walk $X_0 = 0$, $X_{t+1} - X_t \sim f(X_t) \delta_0 + (1-f(X_t))\operatorname{Rademacher}$, where $\delta_0$ is the Dirac delta on ...
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### Probability of filling a small ball before exiting a big one for $d=2$

Let $S_n$ be the simple random walk in dimension $d=2$. Let $0<r<R$ and $\alpha \in (0,1)$. Let $B_r$ denote the $\{x \in \mathbb Z^2: \|x\|\le r\}$ where $\|\cdot\|$ is the Euclidean norm. ...
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### Non linear random walk on 1D lattice

Consider a random walk on a $1$ dimensional lattice, where the probability $q_i(t)$ that the particle is at site $i$ at time $t$ obeys the following equation: \frac{ \mathrm{d}}{ \...
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### Inducing maps between Martin boundaries

This is a reworking of a question I asked on math.se. Given two countable discrete metric spaces $X_{1}$ and $X_{2}$, each equipped with a (irreducible and transient)* random walk given by transition ...
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### A random walk/ruin theory problem with steps whose distribution has infinite mean

In what follows, I will make liberal use of the notations and terminology from ruin theory, just because I think it makes matters more intuitive. However, the problem I'm posing does not depend on its ...
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### References for irrational random walks

I am interested in the symmetric random walk on $\mathbb{R}$ which increments have the discrete law $$\mu=\sum_{i=1}^q p_i (\delta_{\omega_i}+\delta_{-\omega_i})$$ where the $p_i$ sum to $1/2$ and the ...
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### Uniform upper bounds for the return probability of random walks on $\mathbb{R}$

Let $\mu$ be a probability measure with finite support on integers or the real line with the property that $\mu( 0) \le p$ for a fixed $0<p <1$. Let $S_n$ denote the random walk starting at $0$,...
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### Mixing time for random walk on graph with $k$ loops on each vertex

I try to find an upper bound for the mixing time of a random walk $S$ on a connected graph $L=(V,E)$ which has $k<\min_{v\in V}d(v)$ loops at every vertex. The transition probabilities of this ...
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### When do 3D random walks return to their origin?

The probability of a random walk returning to its origin is 1 in two dimensions (2D) but only 34% in three dimensions: This is Pólya's theorem. I have learned that in 2D the condition of returning to ...
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### 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 ...
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### random walk on a bounded surface

apologies in advance for this under formed question. I'm trying to model a random walk process on some 2D surface (like a sphere or cylinder). I have been having some trouble finding literature on ...
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### Random walk with exponential decay

A problem which arises in learning algorithms is $$x_{k+1}= \alpha x_k + \beta e_k$$ where $x_k$ is the scalar state variable at time $k$ and $e_k$ is an independent $\mathrm{Normal}(0,1)$ excitation ...
263 views

### Random walk on infinite graph

Let $G$ be an infinite countable non-oriented connected graph with bounded degrees. Let $X(n)$ be the lazy random walk on $G$ and let $u,v$ be two vertices. Does the ratio $P(X(n)=v)/P(X(n)=u)$ tend ...
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### Arcsine law for gaussian random walk

Consider the random walk $S_i=Z_1+\dots+Z_i$, where the $Z_i$'s are iid $N(0,1)$ and $i\in\{1\dots n\}$. Is there an arcsine law, similar to the one for the $\{\pm 1\}$ case?
### Random process on a sequence of rolls of an $n$-sided die
Let $\ X:=X_{k\,n}\$ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\$ where $\ k\in\{1, 2, \ldots,n\}\$ and $\ n\$ are fixed. Let $t$ be a ...