The random-walk tag has no usage guidance.

**2**

votes

**1**answer

129 views

### How many times does a simple symmetric random walk of length n return to the origin?

Consider the simple symmetric random walk on the integers starting from
the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as
$$ w:= \omega_0 \omega_1 \ldots ...

**0**

votes

**0**answers

45 views

### continuous vs discrete random walk [on hold]

For 1D random walk in discrete case the probability $P_N(X)$ of finding walker at position $X$ after $N$ steps has a binomial distribution, moreover when $N+X$ is odd then probability is 0.
Let's ...

**-1**

votes

**0**answers

8 views

### Hypothesis testing for independent and non-identical distribution [migrated]

I want to apply the hypothesis for my problem. According to A. Wald regarding sequential hypothesis, he used independent and identically distributed (iid) observations or samples but in my case my ...

**0**

votes

**1**answer

108 views

### Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...

**0**

votes

**1**answer

83 views

### Weak convergence of process

Background:
I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist:
Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) ...

**-1**

votes

**1**answer

78 views

### Mixing time of lazy random walk on the directed cycle $C_n$

Briefly: A hint (if this is easy), reference or derivation would be of great help.
The question
Let $C_n$ be the directed cycle with loops in each of its $n$ vertices, and consider the random walk ...

**0**

votes

**0**answers

133 views

### Shortest distance on a combinatorial chess board

Consider a random $n\times n$ combinatorial $0/1$ square matrix over field $\Bbb F$ of rank $r$ with every row distinct and every column distinct as a chess board.
Definition of combinatorial ...

**2**

votes

**1**answer

93 views

### Resources to study self-avoiding walks

What would the best resources be for someone who wants to study self-avoiding walks from a mathematical standpoint?
I'm talking about seminal/important papers, good textbooks perhaps, things of that ...

**1**

vote

**1**answer

162 views

### Does walk on $Z^d$ with steps $(\pm 1,\pm1,\ldots,\pm 1)$ return to origin?

If the steps are iid uniform as in the title, is the return probability known? Is it positive? Answers, comments, references welcome. Clearly each of these steps is not equivalent to $d$ steps of type ...

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votes

**2**answers

312 views

### Adaptive version of the Azuma–Hoeffding inequality

The Azuma inequality states that if we have a martingale $X_1,\ldots,X_N$ that satisfies a bounded difference condition:
$$|X_k - X_{k-1}| \leq c_k$$
Then:
$$\Pr\left[X_N - X_0 \geq \sqrt{2\sum_kc_k^2 ...

**3**

votes

**2**answers

179 views

### Distribution of the hitting time of a random walk

Consider the random walk on $\mathbb R$ with $X_0 = a >0$ and
$$X_{n+1} = X_n + U_n,$$
where $U_0, U_1, U_2,\ldots $ is an i.i.d. sequence of uniform random numbers in $[-1,1]$.
How does the ...

**4**

votes

**1**answer

157 views

### Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) ...

**3**

votes

**0**answers

58 views

### What is the influence of unreliable comparisons on the results of sorting

Considering sorting algorithms based solely on binary comparisons of the elements to be sorted(algorithms such as insertion sort, selection sort, quicksort, and so on), what problems do we face when ...

**2**

votes

**2**answers

168 views

### Bound that random walk stays within with constant probability?

For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$.
What is the bound, in terms of $n$, such that if the ...

**1**

vote

**1**answer

119 views

### Two types of random walkers on square lattice

Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index ...

**1**

vote

**2**answers

72 views

### Position likelihood in a 2D graph [closed]

I am looking for general principles or specific answers to this generic example.
Assume a 2d grid with no boundaries and a roving dot (ant/drunk guy/particle) that is initially located at some ...

**7**

votes

**2**answers

254 views

### Distribution of $\max_{n \ge 0} S_n$, random walk

Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...

**5**

votes

**2**answers

163 views

### Recurrence of Poisson binomial distributed random walk

Let $X_n$ be the outcome of a Bernoulli trial where the probability of getting 1 is $p_n$ and the probability of getting 0 is $1-p_n$, and let $S_n = \sum_{i=1}^n \left(X_i - \textrm{E} X_i \right)$. ...

**0**

votes

**1**answer

127 views

### Discrete random walk with uniformly distributed transition p, set initially

I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk:
Choose $p$ from $U(0,1)$
Start ...

**4**

votes

**3**answers

275 views

### Maximum difference between heads and tails in absolute value

I toss a fair coin $n$ times. Some notation:
$S_i=$ difference between #heads and #number of tails after the first $i$ tosses, $1\leq i\leq n$.
$M_n=\max(S_1,S_2,\dots,S_n)$,
...

**1**

vote

**1**answer

118 views

### A deterministic and explicitly described walk which is like random ones

Consider a sequence $(X_i)_{i = 1}^{\infty}$ which every $X_i$ is $-1$, $0$ or $+1$ and lets define $Y_n = X_1+ \cdots + X_n$. We say the sequence $(X_i)_{i = 1}^{\infty}$ a Good Sequence if $Y_n \neq ...

**9**

votes

**1**answer

338 views

### Lattice random walk under gravity

Suppose a random walk on $\mathbb{Z}^2$ takes a step left or right with
probability $\frac{1}{4}$, but up with probability $\frac{1}{2} p$
and down with probability $\frac{1}{2} (1-p)$, where $p \in ...

**3**

votes

**0**answers

66 views

### Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant?

Is the family of probabilities $\mu^n$ (convolution) generated by a random walk $\mu$ on a finitely generated amenable group $G$ asymptotically invariant ($\|g\mu^n-\mu\|_{L^1}\to 0$ for any $g\in ...

**2**

votes

**1**answer

168 views

### Gradient of Probability Distribution

Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ ...

**3**

votes

**0**answers

161 views

### Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...

**2**

votes

**1**answer

71 views

### More precise statement about lower bounds on the cover time of general graphs

Uriel Feige has shown in 1995 in his paper "A Tight Lower Bound on the Cover Time for Random Walks on Graphs", the following result:
For any graph $G$ on $n$ vertices, and any starting vertex $u$ ...

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votes

**0**answers

71 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. ...

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votes

**0**answers

48 views

### probabilistic interpretation of a finite difference scheme

Let me start with some simple background.
Consider the heat equation :
$
\frac{\partial p}{ \partial t} = \frac{1}{2} \frac{\partial^2 p}{\partial y^2} \quad \mbox{in} \quad \mathbb{R}\times ...

**1**

vote

**1**answer

165 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 $E(S^{2}_{n}| \tau \geq n)$ ...

**1**

vote

**0**answers

100 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 ...

**2**

votes

**1**answer

172 views

### Random walk on a sphere along latitude-longitude grid

Suppose a sphere is partitioned by a latitude-longitude grid, with
grid quadrilaterals $\Delta \times \Delta$.
All grid nodes have degree $4$, while the North & South poles have
degree $2 \pi / ...

**1**

vote

**0**answers

111 views

### Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$

Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.
Let $T_{M}, T_{N}$ be the smallest $n$ such ...

**4**

votes

**2**answers

228 views

### First collision time of $n$ random walkers on a cycle

My question is somehow related to the one here First Collision Time for k Random Walkers on a Torus but, unfortunately, the answer does not cover my concern.
My problem is: consider $n$ walkers on ...

**3**

votes

**1**answer

124 views

### General ballot theorem

I am looking for a version of the Ballot Theorem for general step distributions. Specifically, let $X_1,X_2,\ldots$ be i.i.d. real random variables with some distribution. Let $S_n = S_1 + \cdots + ...

**6**

votes

**2**answers

239 views

### A variant of random walk

Standard random walk assumes a sequence of iid RVs $\{X_i\}_{i\geq 0}$ and studied the distribution of $S_n=\sum_{i=0}^n X_i$.
Here, I am wondering whether there is some work on
$T_n=\sum_{i=0}^n ...

**10**

votes

**0**answers

186 views

### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...

**2**

votes

**1**answer

213 views

### Hitting probabilities for conditioned oriented random walk monotonic?

Consider an oriented random walk on $\mathbb Z^2$ (i.e. only steps $\rightarrow$ and $\uparrow$ with equal probability.) Say we let the walk go $2m$ steps then start guessing sites at distance $2m$ ...

**6**

votes

**2**answers

182 views

### Tail sigma-algebra of a branching random walk

I am looking for any known results about the tail sigma-algebra of a branching random walk. To be specific, let $T$ be the nodes of an infinite binary tree rooted at $r \in T$. Let $\{X_t\})_{t \in ...

**19**

votes

**0**answers

562 views

### Time for Langton's ant to cover a “square” torus

Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel ...

**9**

votes

**5**answers

883 views

### Combinatorial\Probabilistic Proof of Stirling's Approximation

Stirling's approximation is the following well-known asymptotic result:
$$n! \approx \left(\frac{n}{e}\right)^n \sqrt{2 \pi n}$$
This result has several analytical proofs, for example via Laplace's ...

**14**

votes

**1**answer

825 views

### In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = ...

**1**

vote

**1**answer

424 views

### The probability that a 2d continuous time random walk avoids the origin

I am trying to find a reference of a proof of a continuous time version of a result of Dvoretzky and Erdos from their paper "Some problems on random walk in space" that says the probability ...

**0**

votes

**1**answer

75 views

### Stationary distribution of random walk alias solving uncountably many linear equations [closed]

Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$.
Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...

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votes

**0**answers

203 views

### Asymmetric random walk on the line with barriers

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers).
More generally, suppose we fix any ...

**0**

votes

**1**answer

274 views

### Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where ...

**0**

votes

**1**answer

108 views

### Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?

Let $U$ be an open subset of $\mathbb{R}^n$ such that
$\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice''
(for simplicity you can assume piecewise smooth). I also want to allow the ...

**2**

votes

**1**answer

155 views

### Maximizer of random walk with very small drift

This is an extended question based on
Large deviations for maximizer of random walk with drift.
Let $$S_k = X_1 + \ldots + X_k,$$ where $X_i$ are i.i.d. with mean $-\mu < 0$ and unit variance. ...

**5**

votes

**1**answer

339 views

### Properties of a finite random walk

Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise.
Let $Y_N$ be the highest point $X$ have reached on the first ...

**1**

vote

**1**answer

134 views

### Large deviations for maximizer of random walk with drift

Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift?
Let $X_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a ...

**6**

votes

**3**answers

448 views

### Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon

The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%.
This is (part of)
Pólya's theorem.
I have been looking for an analogous (numerical) result for the probability
...