# Questions tagged [random-walk]

For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

2,464 questions
Filter by
Sorted by
Tagged with
37 views

### Is the average of two identically distributed bounded martingales distributed in the same way?

I have seen this question and think it's sufficiently different (and well ranking on google) than the one I'm asking that makes it worth posting. Lets say you have a barrier reflecting martingale, of ...
310 views

### Optimal permutation of transition probabilities in random walk to minimize expected stopping time

Background Consider the set of integers $\{1,\dots,n+1\}$ and a set of probabilities $p_1,\dots, p_n \in(0,1)$. We now define a random walk/Markov chain on these states via the following transition ...
• 6,199
1 vote
81 views

### 2D Modified Random Walk

A particle starts at the point (0,0). It can move +1 to the right with a probability $p$, it can move +1 up with a probability $q$, and it can move diagonally up and to the right with probability $r$. ...
• 1,620
66 views

### Autocorrelation of p-values after $n$ observations

For $i = 1, 2, \dots$, let $X_i \sim N(0, 1)$ and $Y_i \sim N(0, 1)$, with all observations independent of each other. Suppose that after observing $X_1, \dots, X_n, Y_1, \dots, Y_n$ we calculate the ...
• 2,391
54 views

• 162
35 views

### Biased random walk with chance of staying in place

A random walk has probability $x$ of moving $+1$, a probability $y$ of moving $0$, and a probability $z$ of moving $-1$. The PMF for arriving at position $m$ after $n$ steps can be found by summing ...
• 1,620
78 views

### Random walk with indipendent but not identically distributed increments

Suppose $\{Z_i\}_{i=1,2, \ldots}$ are normally distributed (independent but not identically distributed) random variables with mean $\mu(y)>0$ and positive variance $\sigma^2(y)$. Therefore we ...
29 views

### Bounded random walk joint distribution

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. normally distributed random variable with zero mean and variance $\sigma^2$. Consider the bounded random walk $(S_n)_{n\in\mathbb{N}}$, defined ...
• 143
1 vote
52 views

### Mean hitting time of position-dependent random walk

The question I am interested in a continuous-time random walk on $K$ states, numbered $0$ to $(K - 1)$. For each $k$: I call $p_k$ the transition rate from state $k$ to state $(k + 1)$; I call $n_k$ ...
• 373
1 vote
47 views

### Where will a Non-Symmetric Random Walk be after time=t?

I posted this question about ranking the final position of a symmetric random walk after $t$ steps in terms of likelihood: What is the 2nd most likely value of a Random Walk after time=t? I am now ...
• 775
1 vote
54 views

• 81
50 views

### Hitting time and first hitting time

The following statement is in fact a middle step that I want to prove when I try to show that the probability of the first hitting time of symmetric random walk in $\mathbb{R}^{n+1}$ equals $t$ is ...
• 447
18 views

52 views

### Why do three random walkers on $\mathbb Z$ meet infinitely often?

A paper of Dvoretzky and Erdős refers to the fact that three random walkers independently undertaking a simple symmetric random walk on $\mathbb Z$ will meet infinitely often with probability $1$, but ...
• 582
74 views

### Solve for probability of 0 successes where number of attempts is a random variable with unknown distribution

I am interested in solving the following problem: Imagine we have some event A that can occur with probability p, let q = 1 - p. The number of attempts we have for event A to occur is itself a random ...
• 11
101 views

### Transience of random walk on the natural numbers.

Let $X = (X_n)_{n\in\mathbb{N}} \sim \text{MC}(\lambda,P)$ be a Markov chain on the natural numbers such that $\forall i\neq 0,\, (p_{i,i+1}=2/3$ and $p_{i,i-1}=1/3)$ and $p_{0,1}=1$ (every other ...
• 2,504
36 views

### Bounded Random Walks vs Non-Bounded Random Walks

Recently I posted this question here on the probability distribution of two Random Walks (1 dimensional and on the integer line) meeting each other: When and Where will 2 Random Walks Meet for the ...
• 3,196
110 views

### Gambler's ruin with asymmetric probabilities

A gambler with a capital of $1$ dollar goes to a casino, where they participate in a series of games. In each game, regardless of the others, they can win $1$ dollar with probability $p$ or lose $1$ ...
• 1,118
1 vote
101 views

### What Percent of the Time will a Given Random Walk will Behave a Certain Way?

Suppose I have a 1 Dimensional Random Walk (on the integer line) with the following properties (initial conditions): At time=0 it starts at position=0 At each time point, there is a 0.5 probability ...
• 3,196
106 views

### When and Where will 2 Random Walks Meet for the First Time?

This is a question I thought of recently: (Based on some set of initial conditions, i.e. initial positions and movement probabilities , and the current time and positions) When and Where will 2 Random ...
• 3,196