# Questions tagged [random-walks]

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• 65
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### Adjust edge weights via heading on hexagon

I wish for a way to adjust weights based on the edges / directions of a hexagon. If a heading agrees with a certain edge (perpendicular to an edge), then the weights should be distributed as such: ...
• 101
124 views

### Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)

Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows. First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$. ...
• 2,542
231 views

### Local probabilities for lattice random walk

Let $\epsilon <1/2$. Let $X$ be a random variable in $\mathbb Z$ such that $\mathbb P (X=x)\le \epsilon$ for any $x\in \mathbb Z$ (you may add any moment or regularity conditions on $X$ if needed)....
• 723
1 vote
45 views

### Distribution of a random summand conditional on the sum of independent identically distributed copies

The question is simple to state: Let $S_n = \sum_{i=1}^n X_i$, where $X_i$ are independent random variables with a common distribution $F$. For simplicity, assume that $X_i \geq 0$ with prob. 1, and ...
• 115
81 views

### Invariance principle: Brownian bridge and random walk conditioned on end point

Let $\{X_i, i \in \mathbb{N}\}$ be a sequence of non-lattice i.i.d. centered random variables, $\mathbb{E} |X_1| ^3 < 0$. Let $S_n = \sum\limits _{i=1} ^n X_i$ be the corresponding random walk and ...
• 664
286 views

### Average and max. hitting time to a specific vertex

Consider simple random walks that stop when reaching a given node $x$ in an undirected, unweighted and connected graph on $n$ nodes. Let $H(i,x)$ denote the (expected) hitting time from $i$ to $x$, ...
• 63
96 views

77 views

### Non-linear diffusion on networks

The diffusion equation with constant diffusion $D$ can be represented as: \begin{equation} \frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t) \end{equation} where $\Delta$ is the Laplace ...
• 107
918 views

### Expected number of games until bust

The game allows you to bet only \$1 at a time, and if you win, you end up \$1 richer, otherwise you end up \$1 poorer. Probability of winning is$p=0.45$. If you start with \$1 - what is expected ...
41 views

### Recurrence criterion for non-reversible random walks on general infinite (locally finite) graph with unequal edge weights

Can someone please provide a reference (starting point) for analysing recurrence/transience of random walks on graphs with general edge weights? Looking into random walks that are known to be NOT ...
• 15
109 views

### Asymptotics of the return probabilities of a random walk on a transitive graph

Consider a random walk on an infinite connected vertex-transitive graph. Let $f(t)=P_{o,o}^{2t}$ be the probability that the random walk is at its origin at time $2t$. What can be said about the ...
• 927
230 views

### Random walk with decreasing steps

I have a random walk $$R(t)= \sum_{n<t} X_n,$$ with $X_n \sim U(-\tfrac{1}{n^\alpha}, \tfrac{1}{n^\alpha}),$ where $X_n$ are independant and $\alpha >0$. I think that someone must have studied ...
• 85
1 vote
77 views

### Understanding the statements of Theorem 5.5 and Lemmas 5.6, 5.7 and 5.8 from a French paper by Yves Guivarc’h and Émile Le Page

I would like to understand the statement and the proof Theorem 5.5 just for the special case when $X$ is a single point from the paper “Simplicité de spectres de Lyapounov et propriété d’isolation ...
1 vote
121 views

• 65
1 vote
55 views

124 views

### 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 ...
• 19.6k
153 views

### 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 ...
• 363
1 vote
86 views

### 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 ...
• 2,949
1 vote
174 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
144 views

### 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 ...
65 views

### 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. ...
• 406
94 views

### 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 ...
115 views

### 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 $0$,...
• 5,802
98 views

### 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 ...
• 103
1 vote
129 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 ...
• 5,870
345 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 ...
• 723
Let $x(t+1) = x(t) + e(t)$, $e(t)$ iid $\mathcal{N}(0,1)$. What is the probability of $x(s)> c$, for any $0<s<T$? Calculation for any specific $s$ is easy. But I am looking for the ...
Let $P^1$ be the real projective line, identified with lines of $\mathbb{R}^2$. Let $\mu$ be a probability measure on invertible linear map of $\mathbb{R}^2$ and let $(A_i)$ be i.i.d. random variables ...