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

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How to model time changing random variables

Lets say I have a random variable $X(t)$ which describes some unit of motion of a living organism and $X(t)$ is itself a timeseries since this unit of motion changes in time. I would like to be able ...
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47 views

A generalization of simple random walk

Suppose $S_n, n\geq 0$ is a martingale on $\mathbb{R}$ such that $S_0=0$ and $|S_{n+1}-S_{n}|\in [\frac{1}{2}, 1]$. Prove that there exists $c,C>0$ s.t. $$ \frac{c}{\sqrt{n}} \leq P( S_1\geq ...
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38 views

Biased Random Walk with Variable Probability

Consider a random walk in which the probability to move forward in time $t$ is $p_t$ and the probability to move backward is $q_t=1-p_t$ with $p_t<q_t$ with $p_t<p_{t+1}$ and $q_t>q_{t+1}$. ...
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2answers
488 views

Random walk, Cat and mouse

Here is the problem. In graph G, on different vertices there is cat and mouse. Cat and mouse do independent random walk, but time is synchronous, in one unit of time both cat and mouse do one step. ...
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31 views

Random Walk: Position of Particle

Question 2(a) of Chapter 14 of Feller Vol 1: Prove with the notations of section 2: In a random walk starting at the origin find the probability to reach the point a>0 before returning to the origin ...
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1answer
305 views

Expected number of steps for reaching $K$ in a random walk

Assuming steps are $+1/-1$ with a $50/50$ probability. What is the expected step count for reaching $10, 100$ or $K$?
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32 views

Maximum difference between 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, $1\leq i\leq n$. $M_n=\max(S_1,S_2,\dots,S_n)$, ...
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43 views

Is the sudden appearance of transient random walks in 3-dimensions a phase transition?

Consider a particle walking uniformly at random on the infinite d-dimensional lattice $\mathbb{Z}^d$. This is symmetric random walk. Symmetric random walk in two dimensions almost always returns to ...
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73 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
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1answer
369 views

Simple Probability Matrix

Consider a simple model that predicts whether you pass you next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your ...
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48 views

Survival probability of 1D Random Walker [duplicate]

For a 1 D random walk on $Z$ axis, starting at $z=0$, equal probability to go to right or left, what is the probability that during the first k steps the walker's position remains $z\leq m$? This is ...
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72 views

Expected travel of random walk in arbitrary game with multiple payouts

As explained here, the average distance or 'travel' of a random walk with $N$ coin tosses approaches: $$\sqrt{\dfrac{2N}{\pi}}$$ What a beautiful result - who would've thought Pi was involved! ...
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1answer
619 views

Why is the expected average displacement of a random walk of N steps not $\sqrt N$?

Let $D_N$ be the expected average of the displacement of a random walk on $\mathbb Z$ from the origin, where $N$ is the number of steps, each of which is either $-1$ or $1$. We take the definition of ...
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7 views

Limiting Behavior of Zero Mean Random Walk [duplicate]

Suppose $\{ X_t \}$ is a sequence of i.i.d. random variables, with support $\{-1,1\}$ and distribution $P(1)=P(-1)=1/2$. Thus, $S_t = \sum_{s=1}^{t} X_s$ is a zero mean random walk and $$-\infty = ...
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1answer
5 views

Random Walk Limit Behavior

Suppose $\{ X_t \}$ is a sequence of i.i.d. random variables, with support $\{-1,1\}$ and distribution $P(1)=P(-1)=1/2$. Thus, $S_t = \sum_{s=1}^{t} X_s$ is a zero mean random walk. Also, $S_t$ is a ...
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67 views

Probability that the nunber of ties in $2n$ coinflips is $k$

A fair coin is flipped $2n$ times. If the number of "heads" and the number of "tails" coincide, a tie is reached. What is the probability $p_k$, that the number of ties occuring is exactly $k$, ...
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1answer
56 views

Probability, that a sequence of $n$ coin-flips contains $k$ changes of the lead

A fair coin is flipped $n$ times. What is the probability $p_k$, that the lead between "heads" and "tails" changes exactly $k$ times ? For example, the sequence $$HHTTTHH$$ contains two changes ...
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1answer
320 views

Probability of being at a certain point after $N$ steps in Random Walk with a single absorbing barrier

A random walker in $1$ dimension starts walking from a point $k>0$ with an absorbing barrier at point $0$. What is the probability that he will reach a point $m>0$ in $N$ steps? How should I ...
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1answer
31 views

How to prove that $\Bbb{P}(X_{4l} = 0) \leq c_l (2d)^{-2l}$ for some constant $c_l$?

Let $(X_n)$ be a simple random walk on $\Bbb{Z}^d$ starting at $0$. (The dimension $d$ will vary, but I will suppress the dependence on $d$ for brevity.) I encountered a statement which claims that ...
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1answer
43 views

probability distribution for each step in a drunkards walk

Imagine a typical drunkards walk (2D) made of steps $\ell$ each of length $L$ in any direction. I was told that the probability distribution of each step can be written as a Dirac delta like this ...
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2answers
73 views

Limit of an absorbing random walk. (Limit of power of real symmetric matrix)

I have a problem that comes from absorbing random walks on a connected undirected graph $G$ with two types of nodes, absorbing nodes and free nodes. We randomly pick a node to start, once the random ...
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1answer
26 views

Obtaining a $3$-dimensional simple random walk from a $d$-dimensional simple random walk with $d>3$.

Suppose $S_n$ is a $d$-dimensional random walk with $d>3$. Let $T_n=(S_n^{(1)},S_n^{(2)},S_n^{(3)})$, that is, we obtain $T_n$ by looking only at the first three coordinates of $S_n$. It is clear ...
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1answer
112 views

Normalized hit times of a simple RW converge in distribution to hit times of standard Brownian Motion

I would appreciate some hints or guidance towards solving the following exercise: Let $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,\ldots\right\}$ be a simple random walk on the ...
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17 views

Escape probability in diffusion and random walk

I know that in one and two dimension the probability of a random walker to not come back to the origin is zero and in three dimension it is non zero. Is this fact true for diffusion in one, two and ...
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1answer
24 views

Hitting times of a biased continuous time random walk

Let $X_{s \geq 0}$ be a continuous time random walk on $\mathbb{Z}$, i.e. waiting times between jumps are exponentially distributed with mean one. The random walk is biased: $\mathbb{P}(X_s\text{ ...
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1answer
70 views

Reaching a level before another for a random walk

Suppose we are given a simple random walk starting in $0$, i.e. $(X_k)_{k\in\mathbb{N}}$ with $P[X_k=+1]=P[X_k=-1]=\frac{1}{2}$. What is the probability of hitting the level $a$ before hitting the ...
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19 views

Two-dimensional random walk in force field

I have a random walk in two dimensions with a force field. $\vec{x}_{\tau+1} = \vec{x}_\tau + \vec{F}(\vec{x}_\tau)$, where $\vec{x}$ is the position and $\vec{F}(\vec{x}) = ...
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1answer
45 views

Conditional Gambler ruin problem

A gambler repeatedly plays a game where in each round, he wins a dollar with probability 1/3 and loses a dollar with probability 2/3. His strategy is “quit when he is ahead by 2 dollars”, though some ...
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1answer
80 views

Expected Value of a Mosquito

A mosquito is walking at random on the nonnegative number line. She starts at $1$. When she is at $0$, she always takes a step $1$ unit to the right, but, from any positive position on the line, she ...
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What is the area covered by a Random walk in a 2D grid?

I am a biologist and applying for a job, for which I need to solve this question. It is an open book test, where the internet and any other resources are fair game. Here's the question - I'm stuck on ...
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1answer
61 views

Why does symmetry happen in reset-based random walks?

Studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it with ...
3
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2answers
211 views

Exponential ergodicity of biased random walk confined to the positive integer quadrant

I am looking at a discrete-time random walk on $(\mathbb{Z}^{+})^n$, where $\mathbb{Z}^{+}:=\{0,1,2,\dots\}$ and $n\in\mathbb{N}$. At any time, the random walk chooses a uniformly random co-ordinate ...
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3answers
1k views

Exact probability of collision of two independent random walkers after N steps

Two drunks start together at the origin at $t=0$ and every second they move with equal probability either to the right or to the left, each drunk independently from the other. What is the probability ...
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45 views

Expected distance of biased 1d random walk with 0 drift

There were many question about biased 1d random walks earlier, but as far as I can tell none of these are directly related. Let $p,q>0$ such that $p+q=1$. Let $X_0=0$ and $X_{i+1} = X_i+1$ with ...
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21 views

Probability of random walk visit in nonameanable graphs

Consider a vertex-transitive nonameanable graph. Consider a site $x$ having a graph distance $d$ from the origin and let $X(n)$ be a random walk starting from $x$. Is there a general upper bound as a ...
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1answer
305 views

A Boundary crossing result for discrete brownian bridge

Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process $$ ...
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1answer
270 views

How to combine the four Theorems in order to prove the statement?

I have a question concerning a statement about Random Walks on $\mathbb{Z}$. Let $F$ be a distribution on $\mathbb{Z}$ which has mean $0$ and finite variance. Let $\left\{X_1,X_2,\ldots\right\}$ be an ...
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0answers
34 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 $t$ ...
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1answer
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Mean Squared Displacement of Biased Random Walk [closed]

If $x_t=x_{t-1}+\mathcal{N}(\mu,\sigma)$ and $x_0=0$ what's the value of $\langle x_t^2\rangle$?
2
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1answer
73 views

Autocorrelated, discrete, bounded and symmetric random walk with no edge attraction

I need to move over a discrete set of linearly organized.. let's say "Japan steps" $S=\{0,\dots,c\}, c \in \mathbb{N}^*$. My current position is given by $d \in S$. On each time step, I need to draw ...
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1answer
52 views

Expected number of return for a random walk on a graph

Let $G$ be a simple, connected undirected graph of order $n$ and vertex set $\{v_1,\ldots,v_n\}$ and let $P = (p_{i,j})$ be a $n \times n$ matrix where $$p_{i,j} = \left\{ \begin{array}{ll} ...
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3answers
287 views

Probability that after 10,000 steps (+-1) you'll end up at the origin. How to use Central Limit Theorem?

Starting at the origin and taking one step left or right with equal probability, what is the probability that you'll end up at 0 after 10,000 steps? I figured it'd be ...
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2answers
1k views

Random walk on $n$-cycle

For a graph $G$, let $W$ be the (random) vertex occupied at the first time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the ...
3
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1answer
332 views

Random walk around circle [duplicate]

For one of the exercises of my homework I need to answer the following question, but I am not sure how I should apply gamblers ruin theory to solve this problem (it is stated as a hint, not that I ...
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0answers
30 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
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0answers
20 views

Function/algorithm to generate a random walk on a graph

I'm looking for a graph function or an algorithm that can generate a random fluctuating random walk that will eventually converge between the value of y = 0 and y = 1, more or less after a number of ...
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0answers
21 views

Diffusion Constant of a 1D Random Walk

Brownian motion(Wiener process) is a limit of Random walk. What is the diffusion constant for a Brownian motion that is a limit of a 1D Random Walk, with $\frac{1}{2}$ probability of moving to each ...
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1answer
122 views

Expected number of times Random Walk crosses 0 line.

Suppose we have a simple random walk: $$ x_t = x_{t-1} + \epsilon_{t} $$ Where $$ \epsilon_{t} = iid\ \mathcal{N} (0,1) $$ Assume that x starts at ...
2
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1answer
31 views

Derive diffusion coefficient for heat equation from random walk simulation

I want to simulate the underlying stochastic process of diffusion on a microscopic level and compare the result with the solution of the heat equation. However, I'm not able to match the solution of ...
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0answers
18 views

What is the measure for a Random Walk?

Let $F$ be a distribution on $\mathbb{Z}$. Let $(X_1,X_2,...)$ be an i.i.d. sequence of random variables with distribution $F$. Then $S_0=0, S_1=X_1, S_2=X_1+X_2,...$ is called the random walk with ...